and sampling designs for the nonparametric estimation of the regression function in models with correlated errors 93 QFI sntrodution

Trapezoidal rule

he present doumentD resulting of four yers of reserhD would not hve een possile without the guidne of qodD nd the support nd help tht s reeived from mny peopleF s thnk them llF pirst of llD s would like to thnk my supervisorsD n vouhihi nd urim fenhenniD for their ptient guidneD their help nd support during my time s their studentF s hve een so luky to hve hd supervisors who red nd elieved in meF s would lso like to thnk ophie hoExing nd lim fouzed for epting to review my thesisD nd ll the memers of the ommitteeD for their insightful remrks nd their hllenging questionsF e speil thnks to hidier qirrdD for his helpful omments onerning the thesis simultions through these yersF s would like to thnk ll the memers of vortoire ten uuntzmnnD the stu' memers for their helpful ssistneD the tehers the students nd my o0e ollegues for mking my hh life friendly on dily sisF e speil thnks to elin vtour nd gorinne ferzin for their help nd their trust during my tehing lifeF s grtefully knowledge the funding reeived towrds my hh from the elgerin qovernE mentF s would lso like to thnk my professors s hd during my undergrdute nd postgrdute studiesD without whom s would never hd een le to hieve this hhF e speil thnks to ptih wessi nd her wohde forD (rst entrusting me in ttistisD nd ll their help nd their dviesF hnk you for trusting meF htever s syD s n never express my grtitude to my prentsF hnk you so muh for the unonditionl loveD the sri(esD the support nd enourgementsD nd for the pryers you showerd me with ll my lifeF hnk you for the edution you o'ered me leding to this hhF s would like to thnk my husndD for lwys eing there emotionlly nd sienti(llyF hnk you for the time you spent listening to my long omplins nd ll your helpful sienti( dviesF s would lso like to thnk my rotherD my sisters nd my little nieeF por there is no joy they nnot mgnifyD no pin they nnot quell nd no sitution they nnot drw lughter fromF pinllyD thnk you to ll my friendsD who showed me tht no distne n seprte rel ring hertsF hnk you for lwys eing there for meF i Summary sn this thesisD we onsider the (xed design regression model with repeted mesurementsD where the errors form proess with generl utoovrine funtionD iFeF seond order proess @sttionry or nonsttionryAD with nonEdi'erentile ovrine funtion long the digonlF e re interestedD mong other prolemsD in the nonprmetri estimtion of the regression funtion of this modelF e (rst onsider the wellEknown kernel regression estimtor proposed y qsser nd wüllerF e study its symptoti performne when the numer of experimentl units nd the numer of oservtions tend to in(nityF por regulr sequene of designsD we improve the higher rtes of onvergene of the vrine nd the isF e lso prove the symptoti normlity of this estimtor in the se of orrelted errorsF eondD we propose new kernel estimtor of the regression funtion sed on projetion propertyF his estimtor is onstruted through the utoovrine funtion of the errorsD nd spei( funtion elonging to the eproduing uernel rilert pe @urA ssoited to the utoovrine funtionF e study its symptoti performne using the ur propertiesF hese properties llow to otin the optiml onvergene rte of the vrineF e lso prove its symptoti normlityF e show tht this new estimtor hs smller symptoti vrine then the one of qsser nd wüllerF e simultion study is onduted to on(rm this theoretil resultF hirdD we propose new kernel estimtor for the regression funtionF his estimtor is onstruted through the trpezoidl numeril pproximtion of the kernel regression estimtor sed on ontinuous oservtionsF e study its symptoti performneD nd we prove its symptoti normlityF woreoverD this estimtor llow to otin the symptoti optiml smpling design for the estimtion of the regression funtionF e run simultion study to test the performne of the proposed estimtor in (nite smple setD where we see its good performneD in terms of sntegrted wen qured irror @swiAF sn dditionD we show the redution of the swi using the optiml smpling design insted of the uniform design in (nite smple setF pinllyD we onsider n pplition of the regression funtion estimtion in phrmokinetE is prolemsF e propose to use the nonprmetri kernel methodsD for the onentrtionEtime urve estimtionD insted of the lssil prmetri onesF e prove its good performne vi simultion study nd rel dt nlysisF e lso investigte the prolem of estimting the ere nder the onentrtion gurve @egAD where we introdue new kernel estimtorD otined y the integrtion of the regression funtion estimtorF e proveD using simultion studyD tht the proposed estimtors outperform the lssil one in terms of wen qured irrorF he ruil prolem of (nding the optiml smpling design for the eg estimtion is investigted using the qenerlized imulting enneling lgorithmF Keywords. xonprmetri regressionD orrelted oservtionsD utoovrine funtionD reproE duing kernel rilert speD trpezoidl ruleD symptoti normlityD phrmokinetisF Résumé hns ette thèseD nous onsidérons le modèle de régression ve plusieurs unités expérimentlesD où les erreurs forment un proessus d9utoovrine dns un dre générleD 9estEàEdireD un proessus du seond ordre @sttionnire ou non sttionnireA ve une utoovrine non difE férentile le long de l digonleF xous sommes intéressésD entre utresD à l9estimtion non prmétrique de l fontion de régression de e modèleF remièrementD nous onsidérons l9estimteur lssique proposé pr qsser et wüllerF xous étudions ses performnes symptotiques qund le nomre d9unités expérimentles et le nomre d9oservtions tendent vers l9in(niF our un éhntillonnge régulierD nous méliorons les vitesses de onvergene d9ordre supérieur de son iis et de s vrineF xous montrons ussi s normlité symptotique dns le s des erreurs orréléesF heuxièmementD nous proposons un nouvel estimteur à noyu pour l fontion de régresE sionD sé sur une propriété de projetionF get estimteur est onstruit à trvers l fontion d9utoovrine des erreurs et une fontion prtiulière pprtennt à l9ispe de rilert à xoyu eutoreproduisnt @urA ssoié à l fontion d9utoovrineF xous étudions les perE formnes symptotiques de l9estimteur en utilisnt les propriétés de urF ges propriétés nous permettent d9otenir l vitesse optimle de onvergene de l vrine de et estimteurF xous prouvons s normlité symptotiqueD et montrons que s vrine est symptotiquement plus petite que elle de l9estimteur de qsser et wüllerF xous onduisons une étude de simultion pour on(rmer nos résultts théoriquesF roisièmementD nous proposons un nouvel estimteur à noyu pour l fontion de régresE sionF get estimteur est onstruit en utilisnt l règle numérique des trpèzesD pour pproximer l9estimteur sé sur des données ontinuesF xous étudions ussi s performne symptotique et nous montrons s normlité symptotiqueF in outreD et estimteur permet d9otenir le pln d9éhntillonnge optiml pour l9estimtion de l fontion de régressionF ne étude de simultion est onduite (n de tester le omportement de et estimteur dns un pln d9éhntillonnge de tille (nieD en terme d9erreur en moyenne qudrtique intégrée @swiAF he plusD nous montrons l rédution dns l9swi en utilisnt le pln d9éhntillonnge optiml u lieu de l9éhntillonnge uniformeF pinlementD nous onsidérons une pplition de l régression non prmétrique dns le domine phrmoinétiqueF xous proposons l9utilistion de l9estimteur non prmétrique à noyu pour l9estimtion de l fontion de onentrtionF xous véri(ons son on omportement pr des simultions et une nlyse de données réellesF xous investiguons ussi le prolème de l9estimtion de l9eire ous l goure de onentrtion @egAD pour lequel nous proposons un nouvel estimteur à noyuD otenu pr l9intégrtion de l9estimteur à noyu de l fontion de régressionF xous montronsD pr une étude de simultionD que le nouvel estimteur est meilleur que l9estimteur lssique en terme d9erreur en moyenne qudrtiqueF ve prolème ruil de l9otention d9un pln d9éhntillonnge optimle pour l9estimtion de l9eg est disuté en utilE isnt l9lgorithme de reuit simulé générliséF Mots clés. égression non prmétriqueD oservtions orréléesD fontion d9utoovrineD espe de rilert à noyu utoreproduisntD règle des trpèzesD normlité symptotiqueD phrE moinétiqueF Introduction 9

qenerl presenttion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F W yrgniztion of the disserttion F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F II euthor9s ontriutions F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IT e0lition F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IT ulitions F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IT eminrs nd onferenes F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IT rojets F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IT
1 Kernel estimation of the regression function in models with general correlated errors 17

IFI sntrodutionD model nd estimtor F F F F F F F F F F F F F F F F F F F F F F F F F F IV IFP essumptions F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IV IFQ esymptoti results F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IW IFR roofs F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F PI
2 Reproducing kernel Hilbert Space approach in nonparametric regression problems with correlated observations 37

PFI sntrodution F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QV PFP eproduing uernel rilert pes F F F F F F F F F F F F F F F F F F F F F F F F F F RH PFQ gonstrution of the estimtor using the ur pproh F F F F F F F F F F F F F F RQ PFQFI rojetion estimtor F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F RQ PFQFP essumptions nd omments F F F F F F F F F F F F F F F F F F F F F F F F F F RS PFR vol symptoti results F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F RT PFRFI ivlution of the is F F F F F F F F F F F F F F F F F F F F F F F F F F F F F RU PFRFP ivlution of the vrine F F F F F F F F F F F F F F F F F F F F F F F F F F F RV PFRFQ swi nd optiml ndwidth F F F F F F F F F F F F F F F F F F F F F F F F F SH PFRFR esymptoti normlity F F F F F F F F F F F F F F F F F F F F F F F F F F F F F SI PFS gomprison with the qsser nd wüller9s estimtor F F F F F F F F F F F F F F F F F SP PFT imultion study F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F SQ PFU roofs F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F ST PFV eppendix F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F VV iii

List of Figures

PFI he regression funtion of model @wIA is in plin line nd the projetion estimtor is in dshed lineF F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F SR PFP he regression funtion of model @wPA is in plin line nd the projetion estimtor is in dshed lineF

F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F SR
QFI gui regression funtion is in plin line nd the trpezoidl estimtor is in dshed oneF

F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHI
RFI e trjetory of the yrnsteinEhlenek proessF F F F F F F F F F F F F F F F F F F F IQR RFP he oservtions re in lk rossesD the estimtor is in dshed red line nd the onentrtion urve @RFQA is in plin lineF F F F F F F F F F F F F F F F F F F F F F F F IQS RFQ he oservtions in le RFI re in irles nd the estimtor g GM n,h is in dshed lineFIQT RFR gomprison of the optiml designs nd the onventionl design for the estimtion of eg@g oral AF F

F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IRI v
List of Tables PFI he integrted squred isD integrted vrineD swi nd the optiml ndE width for n = 10 nd di'erent vlues of m under the iener error proessD for the qw nd the projetion estimtorsF F F F F F F F F F F F F F F F F F F F F F F F F F F SS PFP he integrted squred isD integrted vrineD swi nd the optiml ndE width for n = 10 nd di'erent vlues of m under the yrnsteinEhlenek error proessD for the qw nd the projetion estimtorsF F F F F F F F F F F F F F F F F F ST QFI he integrted squred isD sntegrted vrineD swi nd the optiml ndE width in terms of m under the iener error proess with σ 2 = 1, for the qw nd the trpezoidl estimtorF F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHR QFP he integrted squred isD sntegrted vrineD swi nd the optiml ndE width in terms of m under the yrnsteinEhlenek error proess with λ = 1 for the qw nd the trpezoidl estimtorF F F F F F F F F F F F F F F F F F F F F F F F F IHR QFQ he integrted squred isD sntegrted vrineD swi nd the optiml ndE width in terms of m under the iener error proess with σ 2 = 0.5 for the qw nd the trpezoidl estimtorF F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHS QFR he integrted squred isD sntegrted vrineD swi nd the optiml ndE width in terms of m under the yrnsteinEhlenek error proess with λ = 25 for the qw nd the trpezoidl estimtorF F F F F F F F F F F F F F F F F F F F F F F F F IHS QFS he integrted squred isD sntegrted vrineD swi nd the optiml ndE width in terms of m under the iener error proess with σ 2 = 0.06 for the qw nd the trpezoidl estimtorF F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHT QFT he integrted squred isD sntegrted vrineD swi nd the optiml ndE width in terms of m under the yrnsteinEhlenek error proess with λ = 50 for the qw nd the trpezoidl estimtorF F F F F F F F F F F F F F F F F F F F F F F F F IHT QFU he swi nd the redutions in the swi of ĝtrap n using the uniform designD theE oretil optiml design nd estimted optiml design when R(s, t) = σ 2 ρ |s λ -t λ |/λ nd n = 5.

F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHU
QFV he swi nd the redutions in the swi of ĝtrap n using the uniform designD theE oretil optiml design nd estimted optiml design when R(s, t) = σ 2 ρ |s λ -t λ |/λ nd n = 10. F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHU QFW he swi nd the redutions in the swi of ĝtrap n using the uniform designD theE oretil optiml design nd estimted optiml design when R(s, t) = σ 2 ρ |s λ -t λ |/λ nd n = 20. F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F IHU QFIH he swi nd the redutions in the swi of ĝtrap n Introduction (français) Présentation générale ves fontions de régression non prmétrique ont été lrgement utilisées dns es dernières déE enniesD ps seulement en ttistiquesD mis dns plusieurs domines tels que l médeine et le tritement du signlF v fontion de régression non prmétriqueD est une fontion générle qui trduit l reltion entre deux vriles X une vrile explitive X et une vrile réponse Y D sns uune restrition prmétrique sur ette fontionF v9un des prolèmes renontrés pr les sttistiiens dns leurs études est l9estimtion de l fontion de régressionD en se snt sur des oservtions prtielles de ette fontionF in ttistiquesD el veut dire qu9on veut estimer l fontion g(•) = E(Y |X = •) à prtir des oservtions (X i , Y i ) 1≤i≤n D qui représentent n opies de (X, Y )F ges oservtions sont souvent modélisées omme suit X Y i = g(X i ) + ε i , où les (ε i ) 1≤i≤n sont des vriles létoires entrées ppelées erreursF hns ette thèseD nous onsidérons le s où les (X i ) 1≤i≤n sont (xées dns un domine et ne sont ps létoires @éhntillonnge déterministeAF g9est le sD pr exempleD des données longitudinles lorsque des unités expérimentles sont oservées à des temps d9éhntillonnge préllement hoisisF r exempleD dns les prolèmes de phrmoinétiqueD les onentrtions d9un ertin médiment dministré dns l9orgnisme sont oservées toutes les demiEheures penE dnt PR heuresF he làD nous onsidérons le modèle de régression non prmétrique dé(ni omme suitX

Y (t i,n ) = g(t i,n ) + ε(t i,n ) pour i = 1, • • • , n, ve 0 ≤ t 1,n < t 2,n < • • • < t n,n ≤ 1F
ve modèle le plus renontré dns l littérture été elui ve des erreurs indépendntesF xous mentionnonsD entre utresD les trvux de riestly et gho @IWUPAD fenedetti @IWUUA et qsser et wüller @IWUWAF ges uteurs ont proposé di'érents estimteurs à noyux pour l fontion de régressionF qsser et al. @IWVRA ont utilisé l9estimteur à noyu dns le s d9une oure de roissne individuelleF pn @IWWPA onsidéré un estimteur linéire lol pondéré dns le s d9oservtions indépendntesF gependntD l9hypothèse que les oservtions sont indépendntes n9est ps toujours rélisteF r exempleD les tilles oservées sur le même enfnt sont orréléesF ves oservtions de tempérture mesurées u ours de l journée sont églement orréléesF our elD nous nous onentrons sur le prolème de l régression non prmétrique où les oservtions sont orréléesF ges modèles ont été utilisés pr plusieurs herheursD prmi lesquels nous itonsD entre utresD rrt et herly @IWVTA qui ont onsidéré des erreurs sttionnires @orréléesAD xúñezE entón et al. et perreir et al. @IWWUA ont exminé une lsse spéi(que d9erreurs non sttionnires I P Introduction (français) ve des fontions d9utovrine prmétriquesD introduites pr xúñezEentón et oodworth @IWWRA pour étudier l9e0ité des implnts ohléiresF our plus de détilsD voir ypsomer et al.

@PHHIA pour une synthèse des résultts sur l régression non prmétrique en présene d9erreurs orrélées sttionniresF in onsidérnt des oservtions orréléesD rrt et herly @IWVTA ont utilisé un estimteur à noyu ien onnu pour l fontion de régressionD proposé pr qsser et wüller @IWUWAF sls ont montré que et estimteur n9est ps onsistntD dns le sens où s vrine ne tend ps vers zéro lorsque le nomre d9oservtions tend vers l9in(niF our elD il fllit onsidérer plusieurs unités expérimentlesD disons mD sur lesquelles n oservtions sont e'etuéesF hns e sD l9estimteur est onsistnt qund m tend vers l9in(niF our des erreurs orrélées non sttionniresD fenhenni et hdi @PHHUA ont ussi onsidéré l9estimteur de qsser et wüllerF ne utre onséquene de l présene de orréltion est l9éhe de plusieurs méthodes de séletion de l fenêtre sées sur les donnéesD telle que l méthode lssique de vlidtion roiséeF our plus de détils sur ette questionD voir pr exempleD eltmn @IWWHAD ghiu @IWVWAD rrt @IWWID IWWRA et ypsomer et al. @PHHIAF hns ette thèseD nous onsidérons le modèle de régression non prmétrique ve des mesures répétéesD dé(nit prX

Y j (t i,n ) = g(t i,n ) + ε j (t i,n ), pour i = 1, • • • , n et j = 1, • • • , m, @IA
où {ε j , j = 1, • • • , m} est une suite de proessus d9erreur entrées indépendntes et identiqueE ment distriuées @iFiFdAF ve l même distriution qu9un proessus εF v non orréltion des oservtions fites entre les unités expérimentles est une hypothèse nturelleD r en générlD les dernières sont hoisies indépendmmentF ge type de données été étudié pr plusieurs uteurs dns le s prmétriqueD nous mentionnons entre utresD ottho' et oy @IWTRA qui ont exmE iné ertines mesures dentires de II (lles et IT grçons à R âges di'érentsD o @IWTSD IWTTAD qrizzle et ellen @IWTWAD qhosh et al. @IWUQA ont onsidéré les méthodes non prmétriques dns les études longitudinlesF xous notons ii que le modèle @IA peut être érit omme suitX

Y (t i,n ) = g(t i,n ) + ε(t i,n ), pour i = 1, • • • , n, où Y (t i,n ) et ε(t i,n
) sont respetivement les moyennes empiriques de {Y j (t i,n ), j = 1, • • • , m} et de {ε j (t i,n ), j = 1, • • • , m}F gel montre que le prolème de l9dpttion de l fontion de régresE sion g à m ouresD pourrit être mené à juster ette dernière à l moyenne de l9éhntillon Y (t i,n )F n utre prolème intéressntD onernnt l9estimtion de l fontion de régression gD est l dérivtion d9un pln d9éhntillonnge optiml {t i,n , i = 1, • • • , n}F ge prolème été lrgement étudié dns le s de l9estimtion pr régression prmétriqueD nous mentionnons les trvux de ks et lvisker @IWTTA et felouni et fenhenni @PHIQAF ges uteurs ont onsidéré e prolème pour une lsse prtiulière de fontions de régressionD et ont otenu le pln d9éhntillonnge optimlF hette et al. @PHIUA ont onstruit une pire d9estimteurs linéires sns iis ve l9éhntillonnge optimle orrespondntD pour omprer deux oures de régression estimées à prtir de deux éhntillons (nis des mesures dépendntesF higljvsky et al. @PHIHA ont proposé une nouvelle pprohe d9éhntillonnge expérimentle pour le modèle de lolistion Q en présene de orréltionY ils ont montré l onvergene file de leur pln d9éhntillonnge vers elui proposé pr fikel et rerzerg @IWUWAF hns le s non prmétriqueD le prolème de l9éhntillonnge optiml été moins développé dns l littértureD en prtiulier lorsque les oservtions sont orréléesF wüller @IWVRA onsidéré un estimteur à noyu et il introduit les points de l9éhntillonnge optimlD lorsque les erreurs sont symptotiquement non orréléesF ge qui n9est ps toujours une hypothèse rélisteD pour les données longitudinles pr exempleF sl utilisé une suite d9éhntillonnge régulier générée pr une fontion de densitéD et il déduit l densité optimle qui minimise l9irreur woyenne udrtique sntégrée @swiA symptotiqueF prwy @IWWHA utilisé l9estimteur de riestly et ghoD et ho et o @PHIPA ont utilisé un estimteur ve une fontionEpoids générleF sls ont onsidéré un éhntillonnge séquentielD pour l9estimtion non prmétrique de l fontion de régression ve des oservtions orréléesF ifromovih @PHHVA onsidéré le prolème de l9éhntillonnge optiml pour une hétérosédstiité onditionnelle en utilisnt l9pprohe pr des séries de pourierF fiedermnn et hette @PHHIA ont proposé une pprohe minimx pour otenir l9éhntillonnge optimlD pr rpport à l9swi symptotiqueD pour un ruit iFiFdF hns plusieurs prolèmes de phrmoinétique @uAD où nous étudions l9tion des médiE ments et leurs utilistions thérpeutiquesD les sienti(ques sont souvent menés à luler l surfe sous l oure de onentrtion @egAF gette mesure représente l9exposition totle de l9orgnisme u médiment dministréF ille est d9intérêt lors du lul de l iodisponiilité du médimentD qui mesure le tux et l9étendue où le médiment tteint le site d9tionD mis el dépend ussi du mode d9dministrtion du médimentF v9eg est l9intégrle de l oure de onentrtion sur l durée d9oservtionF our estimer l9egD deux pprohes di'érentes sont possilesF v9pprohe prmétriqueD où nous estimons les prmètres de l oure de onentrtionD l9eg est lors l9intégrle de l9estimtion plugEin de l oure de onentrtionF gette pprohe été utilisée pr plusieurs uE teursD nous itonsD le livre de hvidin et qiltinn @IWWSAF v deuxième pprohe est l9pprohe non prmétriqueD où l9intégrle eg est estimée diretement sns l9utilistion de l onentrE tionF v méthode l plus utilisée pour l9estimtion de l9eg est l9pproximtion de l9intégrle pr des méthodes qudrturesF v règle des trpèzes est générlement utiliséeD r elle donne une onne estimtion de l9eg lorsque l onentrtion déroît de fçon exponentielleF our plus de détils sur les di'érentes proédures de qudrture numérique et une omprison entre ellesD nous renvoyons le leteur ux trvux de filer et iegorsh @IWWHAF ves temps d9oservtions jount un rôle ruil dns l9e0ité de l9estimtion de l9egD il est intéressnt de trouver le pln d9éhntillonnge optiml pour ette estimtionF lusieurs pprohes pour otenir les points d9éhntillonnge optimux ont été proposées pr plusieurs uE teursF xous mentionnonsD prmi d9utresD utz et h9rgenio @IWVQA qui ont proposé un lgorithme pour minimiser l9erreur moyenne qudrtique @wiA pr rpport ux points d9éhntillonngeF hu'ull et al. @PHHPA ont introduit l9lgorithme de reuit simulé @eA pour otenir l9éhntillonnge optiml et l9ont ompré à d9utres lgorithmes d9optimistionF ves trvux ntérieurs supposent que les oservtions ne sont ps orréléesD e qui n9est ps toujours une hypothèse rélisteF our une fontion de régression linéire prmétrique gD felouni et fenhenni @PHISAD ont inE troduit le pln d9éhntillonnge optiml pour l9estimtion de l9eg lorsque les erreurs sont orréléesF sls ont utilisé l9lgorithme e pour générer l9éhntillonnge optimle et l9ont ompré à l9éhntillonnge uniformeF Organization de la thèse Description courte: hns ette thèseD nous onsidérons le modèle de régression non prmétrique donné pr @IAD ve un proessus d9utoovrine générle sttionnire ou non sttionnireF xous exminons dns un premier temps le prolème de l9estimtion de l fontion de régression gD où nous nlysons l9estimteur à noyu proposé pr qsser et wüller @IWUWAF xous proE posons églement deux nouveux estimteurs à noyu pour l fontion gD à svoir l9estimteur des trpèzes onstruit à prtir de l règle numérique des trpèzesD et l9estimteur de projetion onstruit à l9ide des espes de rilert à noyu reproduisntF xous étudions leurs omporteE ments symptotiques lorsque n et m tendent vers l9in(niD en termes de vitesses de onvergene et de distriutions symptotiquesF xous menons églement une étude de simultion pour tester leurs performnes pour un ensemle (ni d9oservtionsF heuxièmementD nous exminons le prolème de pln d9éhntillonnge optiml pour estimer l fontion de régression gF in(nD nous onsidérons une pplition de l9estimtion de l fontion de régression et de son eg insi que le pln d9éhntillonnge optimlF hns l suiteD nous dérivons le ontenu de l thèse hpitre pr hpitreF Chapitre 1. hns e hpitreD nous onsidérons le prolème de l9estimtion de l fontion de régression g dns le modèle donné pr @IAF xous nous onentrons sur l9estimteur à noyu proposé pr qsser et wüller @IWUWA donnéD pour x ∈ [0, 1]D prX

ĝGM n,h (x) = 1 h n i=1 Y (t i,n ) m i,n m i-1,n K x -t h dt, @PA
où K est un noyu de support [-1, 1]D h = h(n, m) est une fenêtre et les points intermédiires m i,n sont donnés prX m 0,n = 0, m n,n = 1 et m i,n = (t i,n + t i+1,n )/2F hns ette thèseD ontrirement u proessus d9erreur onsidéré dns qsser et wüller @IWUWA et de rrt et herly @IWVTAD nous dptons le modèle d9erreur onsidéré dns fenhenni et hdi @PHHUAF ge proessus d9erreur ε est un proessus du seond ordreD ve une fontion d9utoovrine non di'érentileD tels que les proessus de iener et d9yrnsteinEhlenekF xous onsidérons l suite d9éhntillonnge réguliers {(t i,n ) 1≤i≤n , n ≥ 1}D générée pr une fonE tion de densité f D dé(nie pr ks et lvisker @IWUHA omme suitX

t i,n = F -1 i n pour i = 1, • • • , n, @QA
où F est l fontion de distriution de l fontion de densité f F xous montrons que nous pouvons méliorer les tux de onvergene de l vrine et du iis de ĝGM n,h en utilisnt l suite @QA u lieu d9une suite d9éhntillonnge engendré pr une densité uniformeD omme e fut le s dns fenhenni et hdi @PHHUAF in e'etD nous otenons le tux 1 n 2 h u lieu de 1 n pour le iis @respetivement le tux

1 mn 3 h 2 + 1
n 2 u lieu de 1 mn pour l vrineAD voir roposition IFQFI et roposition IFQFPF xous otenons églement l fenêtre optimle pr rpport à l9swi symptotiqueF he plusD sous des hypothèses lssiquesD nous démontrons l normlité symptotique de l9estimteur ĝGM n,h lorsque n et m tendent vers l9in(niF v9méliortion des tux de onvergene de l vrine et du iis nous été très utile pour une omprison théorique de l performne l9estimteur ĝGM n,h D à notre nouvel estimteur proE posé u ghpitre PF S Chapitre 2. hns e hpitreD nous onsidérons le même modèle de régression que dns le ghpitre IF xous onstruisons un nouvel estimteur de l fontion de régression gD que nous ppelons estimteur de projetionF get estimteurD qui est ussi un estimteur linéire à noyuD est onstruit en utilisnt l9inverse de l mtrie d9utoovrine des oservtionsD que nous supposons onnue et inversileF sl est sé sur une propriété de projetion et il est donné omme suit pour x ∈ [0, 1] @voir hé(nition PFQFIAX

ĝpro n (x) = n i=1 m x,h (t i )Y (t i,n ), @RA oùD en posnt T n = (t i,n ) 1≤i≤n D les poids (m x,h (t i,n )) 1≤i≤n sont déterminés prX m x,h |Tn = f x,h |Tn R -1 |Tn , où f x,h (t) = 1 h 1 0 R(s, t)K x -s h ds, ve f x,h |Tn := (f x,h (t 1,n ), . . . , f x,h (t n,n )) D R |Tn := (R(t i,n , t j,n )) 1≤i,j≤n D R -1 |Tn l9inverse de R |Tn et m x,h |Tn := (m x,h (t 1,n ), . . . , m x,h (t n,n )) F
our ertins proessus d9erreur lssiquesD lorsque l9inverse de l mtrie d9utoovrine R -1 |Tn est nlytiquement onnue @eFgF proessus de ienerD proessus générlisé de iener et proessus d9yrnsteinEhlenek AD nous donnons une expression simpli(ée de l9estimteur proE posé @voir roposition PFQFI et roposition PFQFPAF v onstrution de et estimteur s9inspire des trvux de ks et lvisker @IWTTD IWTVD IWUHAD gependnt leur ontexte est di'érent du nôtreF sls ont onsidéré le modèle linéire prmétrique g(t) = βw(t) où β est un prmètre réel inonnu et w est une fontion onnue pprtennt à l9espe de rilert à noyu utoreproduisnt ssoié à l fontion d9utoovrine du proessus d9erreur εD noté ur @RAF sls ont églement supposé que l mtrie d9utoovrine est onnue et inversileF hns notre sD l9estimteur est onstruit à prtir de l fontion f x,h que nous vons pu démontrer qu9elle pprtient à l9espe ur @RAF n rppel détillé est dédié ux nomreuses tehniques du ur @RA que nous vons utilisées pour otenir nos résultts théoriquesF xous étudions l performne symptotique de l9estimteur proposé lorsque n et m tenE dent vers l9in(niF ves propriétés de ur @RA permettent non seulement d9otenir l9expression symptotique de l vrineD mis églement de trouver le tux de onvergene optiml de l vrine résiduelle de et estimteurD voir roposition PFRFSF xous otenons l fenêtre optimle h * u sens de l9swi symptotiqueD 9est à direD Chapitre 3. hns e hpitreD nous onstruisons un estimteur simple à noyu pour l fontion de régression dns le modèle donné pr @IAF our motiver ette onstrutionD nous onsidérons l9estimteur à noyu de l fontion g sé sur des oservtions ontinues sur l9intervlle

lim n,m→∞ IMSE(h * ) IMSE(h n,m ) ≤ 1, @SA pour une fenêtre h n,m véri(nt X lim n,m→∞ h n,m = 0 et lim n,m→∞
[0, 1] est donnéD pour tout x ∈ [0, 1]D pr ĝ[0,1] (x) = 1 h 1 0 K x -t h Y (t) dt ve Y (t) = 1 m m j=1 Y j (t), @TA pour un noyu K de support [-1, 1] et une fenêtre h = h(n, m)F
xous renvoyons le leteur ux trvux de flnke et fosq @PHHVA ou de hidi et vouni @PHIQA pour plus de détils sur l9estimteur à noyu de l fontion de régression sé sur des oservE tions ontinuesF hns les s prtiquesD où nous n9vons ès qu9à des oservtions disrètesD nous ppliquons l règle numérique des trpèzes pour pproher l9estimteur ontinuD (n de onstruire un nouvel estimteur plus simpleF v9estimteur des trpèzes sé sur les oservE tions (t i,n , Y (t i,n )) 1≤i≤n D où (t i,n ) 1≤i≤n est une suite de pln d9éhntillonnge régulier {T n } n≥1 D générée pr une fontion de densité f D est donné pour x ∈ [0, 1] pr

ĝtrap n (x) = 1 2n N Tn -1 k=1 ϕ x,h f Y (t x,k ) + ϕ x,h f Y (t x,k+1 ) , @UA où t x,1 < • • • < t x,N Tn sont les points de T n dns [x -h, x + h]D ϕ x,h (t) = 1 h K x-t h D K est un noyu de support [-1, 1] et h = h(n, m) est une fenêtre ve 0 < h < 1F
xous étudions les propriétés symptotiques de l9estimteur proposé ĝtrap n D lorsque le nomre d9unités expérimentles m et le nomre d9oservtions n tendent vers l9in(niF he plusD nous démontrons l normlité symptotique de et estimteur et nous déduisons l fenêtre optimle u sens de l9swi symptotiqueD donnée pr @SAF our et estimteur ĝtrap n D nous dérivons le pln d9éhntillonnge symptotiquement optimlD généré pr l fontion de densité f * qui minimise l9swi symptotiqueF our otenir ette fontion f * D nous minimisons le termeD dns l9swiD qui dépend de l fontion de densité de l9éhntillonnge donnée prX

1 0 α(x) f 2 (x) w(x) dx ∆ = Ψ (α,w) (f ),
où α est l fontion de sut de l dérivée du premier ordre de l fontion d9utoovrine R sur l digonleD et w est une fontion de densitéF xous devons lors résoudre le prolème d9optimistion suivntX

f * ∈ argmin f >0, 1 0 f (x)dx=1 Ψ (α,w) (f ).
U ge prolème d9optimistion est résolu pour l densité optimle suivnteX

f * (t) = {α(t)w(t)} 1/3 1 0 {α(s)w(s)} 1/3 ds 1 [0,1] (t).
he plusD nous démontrons que ette densité optimle f * stisfit le ritère d9optimlité minimx donné pr fiedermnn et hette @PHHIA pour des oservtions indépendntesD u sens qu9elle est rouste pr rpport à l misspéi(tion de l fontion d9utoovrine omme suitX

f * ∈ argmin f >0, 1 0 f (t)dt=1 max (α,w)∈Λ Ψ (α,w) (f ), @VA oùD Λ = (α, w) ∈ (C[0, 1]) 2 1 0 α(t)dt < 1 , 1 0 w(s) 1/2 ds 2 < 2 ,
pour 1 > 0 et 2 > 0 (xésF our tester l performne de l9estimteur proposé pour des ensemles (nis d9éhntillons @peE tits n et mAD nous vons mené une vste étude de simultionF xous montrons que l performne de l9estimteur proposé s9méliore en ugmentnt mF xous omprons églement l9estimteur des trpèzes ĝtrap n ve l9estimteur de qsser et wüller ĝGM n pour di'érentes vleurs de n et m et di'érents 4degrés4 de orréltionF xous montrons que les deux estimteurs ont à peu près l même performneD u sens de l9swiF in(nD nous menons une étude de simultion pour montrer l rédution de l9swi lors de l9utilistion du pln d9éhntillonnge optimlD u lieu du pln uniforme dns un ensemle d9éhntillons (nisF our elD nous vons hoisi une grnde lsse de fontions d9utoovrine prmétriquesD pour lesquelles l densité de l9éhntillonnge optimle dépend d9un prmètre inonnuF xous utilisons ensuite l9lgorithme de reuit simulé générlisé @qqA pour estimer e prmètre et nous otenons insi l9éhntillonnge optiml estimé pr l méthode plugEinF ves simultions montrent que les plns d9éhntillonnges optimux théoriques et estimés réduisent d9une mnière signi(tive l9swiF Chapitre 4. hns e hpitreD plusieurs prolèmes de phrmoinétique sont étudiés pour des données orrélées @simulées ou réellesAF xous exminons d9ord le prolème de l9estimtion de l fontion de onentrtion d9un ertin médiment dministré dns l9orgnismeD pour lequel nous proposons d9utiliser l9estimteur non prmétrique à noyu u lieu d9utiliser des méthodes prmétriquesF xous utilisons l9estimteur de qsser et wüller et nous prouvons ses onnes perE formnes à trvers une étude de simultion et une nlyse de données réellesF ves données sont les onentrtions plsmtiques de digoxine près l9dministrtion orle du tritement onsidéré pr gner et tes @IWUQAF insuiteD nous étudions le prolème de l9estimtion de l9egX

AU C(g) = T 0 g(t)dt,
où T est le dernière temps d9oservtionF xous introduisons un nouvel estimteur à noyu qui est l9intégrle de l9estimteur de l fontion de régressionF xous montronsD à l9ide d9une étude de simultionD que l9estimteur proposé est plus performnt que l9estimteur lssique en terme d9erreur moyenne qudrtiqueF in(nD le prolème ruil de trouver le pln d9éhntillonnge optiml pour l9estimtion de l9eg est étudié à l9ide de l9lgorithme qeF Chapitre 5. hns e hpitreD nous terminons e mnusrit pr des onlusions et nous présentons quelques questions ouvertes et perspetives pprues u ours de l préprtion de ette thèseF

General presentation

xonprmetri regression funtions hve een extensively used in the pst dedesD not only in sttistis utD in severl domin suh sD mediine nd signl proessingF he nonprmetri regression funtionD is generl funtion tht trnsltes the reltion etween two vrilesX the explntory vrile X nd the response vrile Y D without ny form or prmetri restritions on this funtionF yne of the situtions tht sttistiins enounter in their studies is the estimE tion of the regression funtionD sed on prtil oservtions of this funtionF sn sttistil termsD one wnts to estimte the funtion g(

•) = E(Y |X = •) sed on the oservtions (X i , Y i ) 1≤i≤n D whih re n opies of (X, Y )F hese oservtions re often modeled s followsX Y i = g(X i ) + ε i ,
where (ε i ) 1≤i≤n re entered rndom vrilesD lled errorsF sn this thesisD we onsider the se where (X i ) 1≤i≤n re (xed within some domin nd not rndom @(xed designAF his is the seD for instneD of the longitudinl dt when experimentl units re oserved through smpling time pointsD hosen prior to the experimentF es life time exmpleD in phrmokinetis prolemsD the onentrtions of some drug dministrted in the orgnism re oserved every hlf n hour during PR hoursF reneD we onsider the soElled (xed design regression model de(ned s followsX

Y (t i,n ) = g(t i,n ) + ε(t i,n ) for i = 1, • • • , n, with 0 ≤ t 1,n < t 2,n < • • • < t n,n ≤ 1F
he most thoroughly disussed model hs een the one with independent errorsF e mentionD mong othersD the works of riestly nd gho @IWUPA D fenedetti @IWUUA nd qsser nd wüller @IWUWAF he previous uthors proposed di'erent kernel regression estimtorsF qsser et al. @IWVRA used the kernel estimtor in the se of n individul growth urveF pn @IWWPA onsidered weighted lol liner regression in the se of independent oservtionsF roweverD onsidering tht the oservtions re independent is not lwys relisti ssumptionF por instneD the heights oserved on the sme hild re orreltedF he temperture oservtions mesured long the dy re lso orreltedF por thisD we fous on the nonprmetri regression estimtion prolem where the oservtions re orreltedF hese models were used y severl uthorsD we mention mong othersD the work of rrt nd herly @IWVTA who onsidered sttionry @orreltedA errorsD xúñezEentón et al. nd perreir et al. @IWWUA onsidered spei( lss of nonsttionry errors with prmetri utoovrine funtionsD introdued y xúñezE entón nd oodworth @IWWRA to study the e0y of ohler implntsF por review on the nonprmetri regression in the presene of sttionry orrelted errorsD see ypsomer et al. @PHHIAF W IH hen onsidering orrelted oservtionsD rrt nd herly @IWVTA used wellEknown kernel regression estimtor proposed y qsser nd wüller @IWUWAF hey showed tht the kernel estiE mtor is not onsistentD in the sense thtD its vrine does not tend to zero when the numer of oservtions n tends to in(nityF por thisD it ws neessry to onsider severl experimentl unitsD sy mD on whih n oservtions re tkenY then the estimtor is onsistent when m tends to inE (nityF por nonsttionry orrelted errorsD fenhenni nd hdi @PHHUA onsidered the qsser nd wüller estimtorF enother onsequene of the presene of orreltionD is the rekdown of severl dt sed ndwidth seletion methodD suh s the lssil rossEvlidtion riterionF por detils on this issueD see for instneD eltmn @IWWHAD ghiu @IWVWAD rrt @IWWID IWWRA nd ypsomer et al. @PHHIAF sn this thesisD nd for this type of dtD we onsider the soElled (xed design regression model with repeted mesurementsD given s followsX

Y j (t i,n ) = g(t i,n ) + ε j (t i,n ), for i = 1, • • • , n nd j = 1, • • • , m, @IA
where {ε j , j = 1, • • • , m} is sequene of iFiFdF entered error proesses with the sme distriution s proess εF he non orreltion of the oservtions mde on di'erent experimentl units is nturl ssumptionD sine in generlD the lter re hosen independentlyF his type of dt ws investigted y severl uthors in the prmetri situtionD we mention mong othersD ottho' nd oy @IWTRA who onsidered some dentl mesurements of II girls nd IT oys t R di'erent gesD o @IWTSD IWTTAD qrizzle nd ellen @IWTWAD qhosh et al. @IWUQA onsidered the nonprmetri methods in longitudinl studiesF st should e noted hereD tht wodel @IA n e written s followsX

Y (t i,n ) = g(t i,n ) + ε(t i,n ), for i = 1, • • • , n,
where Y (t i,n ) nd ε(t i,n ) re respetively the smple men of {Y j (t i,n ), j = 1, • • • , m} nd {ε j (t i,n ), j = 1, • • • , m}F his shows tht the prolem of (tting the regression funtion g to m urvesD ould e rought to (tting it to the smple mens Y (t i,n )F enother interesting prolem onerning the estimtion of the regression funtion gD is the derivtion the optiml smpling design {t i,n , i = 1, • • • , n}F his prolem hs een extensively studied in the se of the prmetri regression estimtion prolemsD we mention the works of ks nd lvisker @IWTTA nd felouni nd fenhenni @PHIQAF hese uthors onsidered this prolem for prtiulr lss of regression funtionsD nd otined the optiml smpling designF hette et al. @PHITDA onstruted pir of liner unised estimtors with orresponding optiml designs to ompre two regression urves estimted from two (nite smples of dependent mesurementsF higljvsky et al. @PHIHA proposed new pproh of designing experiments for the lotion model in the presene of orreltionY they showed the wekly onvergene of their design to the one proposed y fikel nd rerzerg @IWUWAF sn the nonprmetri seD the prolem of the optiml design ws less developped in the litertureD espeilly when the oservtions re orreltedF wüller @IWVRA onsidered the kernel regression estimtorD nd introdued the optiml design points when the errors re symptotilly unorreltedD whih is not lwys relisti ssumption for the longitudinl dt for instneF re used regulr design sequene generted y density funtionD nd derived the optiml density tht minimizes the symptoti sntegrted wen qured irror @swiAF prwy @IWWHA used the riestly nd gho estimtorD nd ho nd o @PHIPA used generl weight funtion estimtorF hey onsidered sequentil design for the estimtion of the regression funtionD when the oservtions re independentF ifromovih @PHHVA onsidered the prolem of optiml design for onditionl heterosedstiity using the pourier series pprohF fiedermnn nd hette @PHHIA proposed minimx pproh to otin the optiml design with respet to the symptoti swiD for iFiFdF noiseF sn severl prolems of phrmkokinetis @uAD where we study the tion of drugs nd their therpeuti useD sientists re often rought to lulte the ere nder the onentrtion gurve @egAF his mesure represents the totl exposure of the ody to the dministrted drugF st is of interest when lulting the iovilility of the drugD whih mesures the rte nd extent to whih drug rehes the site of tionD ut it lso depends on the wy of dministrtion of the drugF he eg is the integrl of the onentrtion urve over the oservtion timeF sn order to estimte the egD two di'erent pprohes re possileF he prmetri pE prohD where we estimte the prmeters of the onentrtion urveD the eg estimtion is thn the integrl of the plugEin estimtion of the onentrtion urveF his pproh ws used y severl uthorsD we mention mong othersD the ook of hvidin nd qiltinn @IWWSAF he seond pproh is the nonprmetri pprohD where the eg is estimted diretly without the use of the onentrtionF he most used method for the estimtion of the egD is the pproxE imtion of the integrl y qudrture methodsF he trpezoidl rule is ommonly usedD sine it gives good eg estimtion when the onentrtion dereses exponentillyF por more detils on the di'erent xewtonEgotes numeril qudrture proedures nd omprison etween themD we refer the reder to the work of filer nd iegorsh @IWWHAF ine the oservtion times ply ruul role in the e0ieny of the eg estimtionD it is interesting to (nd the optiml smpling times for this estimtionF everl pprohes to oE tin the optiml smpling points were proposed y severl uthorsF e mentionD mong othersD utz nd h9rgenio @IWVQAD who proposed n lgorithm to minimize the wi with respet to the smpling pointsF hu'ull et al. @PHHPA introdued the imulted enneling lgorithm @eA to otin the optiml designD nd ompred it to severl other optimiztion lgorithmsF he previous works suppose tht the oservtions re unorreltedD whih is not lwys relisti ssumptionF por prmetri liner regression funtion gD felouni nd fenhenni @PHISAD inE trodued the optiml smpling design for the eg estimtion when the errors re orreltedF hey used the e lgorithm to generte the optiml design nd they ompred it to the uniform designF Organization of the dissertation Short description: sn this thesisD we onsider the nonprmetri regression model given y @IAD with generl orreltion error proess whih mye sttionry or nonsttionryF e onsider (rstD the prolem of estimting the regression funtion gD where we nlyse the wellEknown kernel regression estimtor proposed y qsser nd wüller @IWUWAF e lso propose two new kernel estimtors for the funtion gD nmely the trpezoidl estimtor onstruted from numeril ruleD nd the projetion estimtor onstruted using the eproduing uernel rilert spesF e study their symptoti ehviors when n nd m tend to in(nityD in terms of rtes of onvergene nd symptoti distriutionsF e lso ondut simultion study to test their performnes for (nite set of oservtionsF eondD we onsider the prolem of (nding the optiml smpling design for estimting the regression funtion gF pinllyD we onsider n pplition of the regression funtion estimtion nd its eg long with the optiml smpling designF sn the sequelD we desrie the ontent of the thesis hpter y hpterF Chapter 1. sn this hpterD we onsider the prolem of estimting the regression funtion g in the model given y @IAF e onsider the wellEknown kernel estimtor proposed y qsser nd wüller @IWUWA givenD for x ∈ [0, 1]D yX

ĝGM n,h (x) = 1 h n i=1 Y (t i,n ) m i,n m i-1,n K x -t h dt, @PA
where

K is kernel of support [-1, 1]D h = h(n, m) is ndwidth nd the midpoints m i,n re given yX m 0,n = 0, m n,n = 1 nd m i,n = (t i,n + t i+1,n )/2F
rereD the error proess is di'erent from tht in qsser nd wüller @IWUWA nd rrt nd herly @IWVTA D ut it is s in fenhenni nd hdi @PHHUAD iFeF ε is seond order proessD with non di'erentile ovrine funtionD suh s the iener proess nd the yrnstein hlenek proessF e onsider the regulr sequene of designs {(t i,n ) 1≤i≤n , n ≥ 1}D generted y density funtion f D whih ws de(ned y ks nd lvisker @IWUHA s followsX

t i,n = F -1 i n for i = 1, • • • , n, @QA
where F is the distriution funtion of the density funtion f F e show thtD we n improve the rtes of onvergene of the vrine nd the is of ĝGM n,h when using @QAD insted of other sequene of designsD s it ws the se in fenhenni nd hdi @PHHUAF sn ftD we otin the rte 1 n 2 h insted of 1 n for the is @respetively the rte 1 mn 3 h 2 + 1 n 2 insted of 1 mn for the vrineAD see ropositions @IFQFIA nd @IFQFPAF e lso derive the optiml ndwidth with respet to the symptoti swiF sn dditionD we prove the symptoti normlity of the estimtor ĝGM n,h when n nd m tend to in(nityD under lssil ssumptionsF he improvement of the rtes of onvergene of the vrine nd the isD were very useful for deriving theoretil omprison of the estimtor ĝGM n,h D to our new proposed estimtorD given in ghpter PF Chapter 2. sn this hpterD we onsider the sme regression model s in ghpter IF e onstrut new estimtor of the regression funtion gF his estimtorD whih is lso liner kernel estimtorD is onstruted using the inverse of the utoovrine mtrix of the oservtionsD tht we ssume known nd invertileF st is sed on projetion property nd is givenD for

x ∈ [0, 1]D s follows @see he(nition @PFQFIAAX ĝpro n (x) = n i=1 m x,h (t i )Y (t i,n ), @RA
where the weights (m x,h (t i,n )) 1≤i≤n re eing determinedD letting

T n = (t i,n ) 1≤i≤n D yX m x,h |Tn = f x,h |Tn R -1 |Tn , where f x,h (t) = 1 h 1 0 R(s, t)K x -s h ds, with f x,h |Tn := (f x,h (t 1,n ), . . . , f x,h (t n,n )) D R |Tn := (R(t i,n , t j,n )) 1≤i,j≤n D R -1 |Tn the inverse of R |Tn nd m x,h |Tn := (m x,h (t 1,n ), . . . , m x,h (t n,n )) F
por some lssil error proessesD when the inverse of the utoovrine mtrix is nlytiE lly knownD we give simpli(ed expression of the proposed estimtorD suh s for the iener proessD the generlized iener nd the yrnsteinEhlenek proess @see ropositions @PFQFIA nd @PFQFPAAF his estimtor ws inspired y the work of ks nd lvisker @IWTTD IWTVD IWUHA ut there ontext is di'erent thn oursF hey onsidered the prmetri liner model with g(t) = βw(t) where β is n unknown rel prmeter nd w is known funtion elonging to the eproduing uernel rilert pe ssoited to the utoovrine funtion of the error proess εD denoted y ur@RAF hey lso ssumed tht the utoovrine mtrix is known nd invertileF sn our se the funtionD through whih the estimtor is onstrutedD f x,h is proven to elong to ur@RAF e detiled rell is dedited to the mny tehniques of the ur@RA tht we used to otin our theoretil resultsF e investigte the symptoti performne of the proposed estimtorD when n nd m tend to in(nityF he properties of the ur@RA llow not only to otin the symptoti expression of the vrineD ut lso to (nd the optiml rte of onvergene of the residul vrine of this estimtor @see roposition @PFRFSAAF e derive the optiml ndwidth h * with respet to the symptoti swiD optimlity in the sense thtD

lim n,m→∞ IMSE(h * ) IMSE(h n,m ) ≤ 1, @SA
for ny sequene of ndwidths h n,m verifyingX lim is n pproximtion of n integrlD nd the est liner pproximtion of n integrl is sed on some projetion propertyD see for instne fenhenni nd gmnis @IWWPAF pinllyD we ondut simultion study in order to investigte the performne of the proposed estimtor ĝpro n in (nite smple setD where we prove its good performne for smll smple sizesF xext we ompre it with the qsser nd wüller9s estimtor ĝGM n,h for di'erent vlues of the numer of experimentl units m nd di'erent vlues of the smple size nF his simultion on(rms our theoretil resultsF Chapter 3. sn this hpterD we onstrut simple kernel estimtor for the regression funtion in the model given y @IAF o motivte this onstrutionD we onsider the kernel estimtor of g sed on ontinuous oservtions on [0, 1]D given for ny

x ∈ [0, 1] yD ĝ[0,1] (x) = 1 h 1 0 K x -t h Y (t) dt with Y (t) = 1 m m j=1 Y j (t), @TA for kernel K of support [-1, 1] nd ndwidth h = h(n, m
)F e refer the reder to the works of flnke nd fosq @PHHVA or hidi nd vouni @PHIQA for more detils on the kernel estimtion of the regression funtion sed on ontinuous oservtionsF sn prtil sesD where we only hve ess to disrete oservtionsD we pply the trpezoidl rule to pproximte the ontinuous estimtor to onstrut new simpler estimtorF he trpezoidl estimtor sed on the oservtions (t i,n , Y (t i,n )) 1≤i≤n where (t i,n ) 1≤i≤n is regulr sequene of designs generted y some density funtion f is givenD for x ∈ [0, 1]D y

ĝtrap n (x) = 1 2n N Tn -1 k=1 ϕ x,h f Y (t x,k ) + ϕ x,h f Y (t x,k+1 ) , @UA where t x,1 < • • • < t x,N Tn re the points of T n in [x -h, x + h]D ϕ x,h (t) = 1 h K x-t h D K is kernel of support [-1, 1] nd h = h(n, m) is ndwidth with 0 < h < 1F
e investigte the symptoti properties of the proposed estimtor ĝtrap n D when oth the numer of experimentl units m nd the numer of oservtions n tend to in(nityF sn dditionD we prove the symptoti normlity of this estimtor nd we derive the optiml ndwidth with respet to the symptoti swiD s given y @SAF por this estimtorD ĝtrap n D we derive the symptoti optiml smpling designD generted y the density funtion f * tht minimizes the symptoti swiF o otin this funtion f * D we minimize the term of swi depending on the design density given yX

1 0 α(x) f 2 (x) w(x) dx ∆ = Ψ (α,w) (f ),
where α is the jump funtion of the (rst order derivtive of the utoovrine R t the digonlD nd w is ny density funtionF e hve then to solve the following optimiztion prolemX

f * ∈ argmin f >0, 1 0 f (x)dx=1 Ψ (α,w) (f ).
his optimiztion prolem is solved for the following optiml design densityX

f * (t) = {α(t)w(t)} 1/3 1 0 {α(s)w(s)} 1/3 ds 1 [0,1] (t).
woreoverD we prove tht this optiml density f * stis(es minimx optimlity riterion given y fiedermnn nd hette for the independent oservtionsD in the sense tht it is roust with respet to the misspei(tion of the error9s utoovrine funtion s followsX

f * ∈ argmin f >0, 1 0 f (t)dt=1 max (α,w)∈Λ Ψ (α,w) (f ), @VA whereD for (xed 1 > 0 nd 2 > 0D Λ = (α, w) ∈ (C[0, 1]) 2 1 0 α(t)dt < 1 , 1 0 w(s) 1/2 ds 2 < 2 .
o test the performne of the proposed estimtor in (nite smple sets @smll n nd mAD we ondut n extensive simultion studyF e show tht the performne of the proposed estimtor gets etter s m inresesF e lso ompre the trpezoidl estimtor ĝtrap n with the qsser nd wüller estimtor ĝGM n for di'erent vlues of n nd m nd di'erent 4degree4 of orreltionF e show thtD oth of the estimtors hve pproximtely the sme performneD with respet to the swiF pinllyD we run simultion study to show the redution of the ext swi when using the optiml smpling designD insted of the uniform design in (nite smple setF por thisD we hose lrge lss of prmetri utoovrine funtionsD where the optiml design density depends on the utoovrine unknown prmeterF e then use the qenerlized imulted enneling lgorithm @qeA to estimte the prmeter nd so we otin the plugin estimted optiml deE signF he simultions show tht oth the theoretil nd the estimted optiml design redue the swi signi(ntlyF Chapter 4. sn this hpterD severl phrmokinetis prolems re investigtedD for orrelted dt @simulted or relAF e (rst onsider the prolem of estimting the onentrtion funE tion of some dministrted drugD where we propose to use the nonprmetri kernel estimtor insted of the prmetri methodsF e use the qsser nd wüller estimtorD nd we prove its good performne vi simultion study nd rel dt nlysisF he dt re digoxin plsm onE entrtions fter n orl dministrtion of tretmentD onsidered y gner nd tes @IWUQAF eondD we investigte the prolem of estimting the egX

AU C(g) = T 0 g(t)dt,
where T is the sientist9s lst smpling timeF e introdue new kernel estimtorD whih is the integrtion of the regression funtion estimtorF e proveD using simultion studyD tht the proposed estimtors outperform the lssil one in terms of wen qured irrorF pinllyD the ruil prolem of (nding the optiml smpling design for the eg estimtion is investigted using the qe lgorithmF Chapter 5. sn this hpterD we give onlusion of the thesis nd we present some opened questions nd perspetives whih ppered during the preprtion of this thesisF IT Author's contributions Aliation

Y j (t i ) = g(t i ) + ε j (t i ), @IFIA
where g is the unknown regression funtion on [0, 1] nd {ε j (t), t ∈ [0, 1]} j is sequene of error proessesF e ssume tht g ∈ C 2 ([0, 1]) nd tht (ε j ) j re iFiFdF proesses with the sme distriution s entered seond order proess εD of utoovrine funtion RF he kernel estimtorD whih will underlie the disussion in this hpterD is proposed y qsser nd wüller @IWUWA nd is givenD for ny

x ∈ [0, 1]D yD ĝGM n (x) = n i=1 Y (t i ) m i m i-1 ϕ x,h (s) ds, @IFPA
where

Y (t i ) = 1 m m j=1 Y j (t i )D ϕ x,h (t) = 1 h K( x-t h )D K is (rst order kernel of support [-1, 1]D h is ndwidth (0 < h < 1) nd (m i ) 1≤i≤n is sequene of midpoints de(ned s followsD m 0 = 0, m i = t i + t i+1 2 for i = 1, . . . , n -1 nd m n = 1. @IFQA
sn this hpterD we investigte the symptoti ehvior of the estimtor ĝGM n when n nd m tend to in(nityF e derive higher order rtes of onvergene for the is nd the vrineD in the se of the soElled regulr sequene of designs (t i,n ) n≥1 D de(ned y ks nd lvisker @IWUHA @see he(nition IFIFIAY hn the ones otined y fenhenni nd hdi @PHHUAD in the se of niform design nd orrelted oservtionsF Denition 1.1.1 Let F be a distribution function of some density f , with inf

t∈[0,1] f (t) > 0 and sup t∈[0,1] f (t) < ∞.
The so-called regular sequence of designs generated by a density f is dened by,

T n = t i,n = F -1 i n , i = 1, . . . , n for n ≥ 1.

Assumptions

sn order to derive the symptoti resultsD the following ssumptions on the utoovrine funE tion R nd the kernel K re requiredF @eA he utoovrine funtion R exists nd is ontinuous on the squre [0, 1] 2 .

@fA et the digonl @iFeF when t = s in the unit squreAD R hs ontinuous left nd right (rstEorder derivtivesD tht isX

R (0,1) (t, t -) = lim s↑t ∂R(t, s) ∂s nd R (0,1) (t, t + ) = lim s↓t ∂R(t, s) ∂s .
he jump funtion long the digonl α(t) ∆ = R (0,1) (t, t -) -R (0,1) (t, t + ) is ssumed to e ontinuous nd not identilly equl to zeroF 1.3. ASYMPTOTIC RESULTS IW @gA y' the digonl @iFeF when t = s in the unit squreAD R is ssumed to hve vipshitz mixed prtil derivtives up to order two whih stis(esX

A (i,j) ∆ = sup 0≤t =s≤1
|R (i,j) (t, s)| < ∞ for all integers i, j such that 0 ≤ i + j ≤ 2.

he previous ssumptions re lssil nd were used in severl worksD see for instneD the works of ks nd lvisker @IWTTD IWTVD IWUHA nd fenhenni nd hdi @PHHUAF @hA he uernel K is of support

[-1, 1]D t lest in C 2 ([-1, 1])D even nd the seond derivtive K is vipshitzF
Example 1.2.1 Examples of processes with autocovariances satisfying Assumptions (A), (B) and (C) are given as follows:

1. The Wiener process with autocovariance function R(s, t) = σ 2 min(s, t), has a constant jump function α(t) = σ 2 and R (i,j) (s, t) = 0 for all i, j such that i + j = 2 and s = t.

2. The Ornstein-Uhlenbeck process with a stationary autcovariance R(s, t) = σ 2 e (-λ|s-t|) for σ > 0 and λ > 0. For this process α(t) = 2σ 2 λ and R (0,2) (s, t) = σ 2 λ 2 e (-λ|s-t|) .

3. A generalization of the Ornstein-Uhlenbeck process to a process with a nonstationary autocovariance function of the form: R(s, t) = σ 2 ρ |s λ -t λ |/λ for σ > 0, λ > 0 and 0 < ρ < 1.

For this process, the jump function is not constant and given by α(t) = -2σ 2 ln(ρ)t λ-1 .

4. Sacks and Ylvisaker (1966) gave another general class of convex stationary autcovariance functions of the form,

R(s, t) = 1/|t-s| 0 (1 -µ|t -s|)p(µ) dµ,
where p is a probability density and p , its derivative, are such that,

lim µ→∞ µ 3 p(µ) < ∞, and ∞ a (µp (µ) + 3p(µ)) 2 )µ 6 dµ < ∞,
for some nite constant a. For this autocovariance function,

α(t) = 2 ∞ 0 µp(µ) dµ for all t.
Example 1.2.2 Example of kernels satisfying Assumption (D) are given as follows:

1. The Quadratic kernel dened by

K(u) = 15 16 (1 -u 2 ) 2 1 {|u|≤1} . 2. The Triweight kernel dened by K(u) = 35 32 (1 -u 2 ) 3 1 {|u|≤1} .
E (ĝ GM n (x)) -g(x) = B 2 h 2 g (x) + o(h 2 ) + O 1 n 2 h , where B = 1 -1 t 2 K(t)dt.
Proposition 1.3.2 Suppose that Assumptions (A)-(D) are satised and that

f ∈ C 2 ([-1, 1]). If lim n→∞ h = 0 and lim n→∞ nh = ∞ then, for any x ∈]0, 1[, Var ĝGM n (x) = 1 m R(x, x) - 1 2 α(x)C K h + o h m + O 1 mn 3 h 2 + 1 mn 2 , where C K = 1 -1 1 -1 |u -v|K(u)K(v)dudv.
e omprison of the previous propositions to heorem P of fenhenni nd hdi @PHHUA yields thtX the rte of onvergene of the is @respetively the vrineA of the qsser nd wüller estimtor to its limitD n e improved when using regulr sequene of designsF ht isD we otin the rte 1 n 2 h insted of 1 n for the is @respetively the rte 1 mn 3 h 2 + 1 n 2 insted of 1 mn for the vrineAF ropositions IFQFI nd IFQFP llow to derive the symptoti expression of the men squred error @wiA of the estimtor @IFPAF he integrted men squred error @swiA is then otined y integrting the wi with respet to weight funtion wF he results re nnouned without proof in the following theoremD sine it is trivil onsequene of the two propositionsF Theorem 1.3.1 If all the assumptions of Propositions 1.3.1 and 1.3.2 are satised then for any

x ∈]0, 1[, MSE(ĝ GM n (x)) = 1 m R(x, x) - 1 2 α(x)C K h + 1 4 h 4 (g (x)) 2 B 2 + o h 4 + h m + O h n 2 + 1 n 4 h 2 + 1 mn 3 h 2 + 1 mn 2 . IMSE(ĝ GM n ) = 1 m 1 0 R(x, x)w(x) dx - C K h 2m 1 0 α(x)w(x) dx + B 2 4 h 4 1 0 [g (x)] 2 w(x) dx + o h 4 + h m + O h n 2 + 1 n 4 h 2 + 1 mn 3 h 2 + 1 mn 2 ,
where w is a continuous density function, B and C K are dened in the two propositions above.

he symptoti optiml ndwidth is otined y minimizing the symptoti swi nd is given y the following propositionF Proposition 1.3.3 (Optimal bandwidth) Suppose that the assumptions of Theorem 1.3.1 are satised. Moreover assume that m n = O(1) as n, m → ∞. Denote by IMSE(h) the IMSE of the Gasser and Müller estimator when the bandwidth h is used. Then the bandwidth,

h * = C K 1 0 α(x)w(x) dx 2B 2 1 0 [g (x)] 2 w(x) dx 1/3 m -1/3 , @IFRA
√ m ĝGM n (x) -g(x) D -→ Z, with Z ∼ N (0, R(x, x)),
where D denotes the convergence in distribution and N is the normal distribution.

Proofs

Proof of Proposition 1.3.1.

vet T n = {t i,n , 1 ≤ i ≤ n} @for the ske of lrityD we shll omit the n in t i,n when there is no miguityAF ine E(Y (t i )) = g(t i ) for i = 1, . . . , n then for ny x ∈]0, 1[ we hveD

E(ĝ GM n (x)) = n i=1 g(t i ) m i m i-1 ϕ x,h (s) ds. vet N Tn = Card I x,h = Card{i = 1, • • • , n/[m i-1 , m i ]∩]x -h, x + h[ = ∅}F
he de(nition of (m i ) 1≤i≤n yields tht N Tn ≥ 1F vet t x,i e the points of T n for whih i ∈ I x,h F por h smll enough nd without loss of generlityD we tke the following nottionX

0 < m x,0 < x -h ≤ m x,1 < • • • < x + h ≤ m x,N Tn < 1. @IFSA husD sine ϕ x,h (s) = 0 for s / ∈]x -h, x + h[ thenD E(ĝ GM n (x)) = N Tn i=1 g(t x,i ) m x,i m x,i-1 ϕ x,h (s) ds. vetD l h (x) ∆ = x+h x-h ϕ x,h (s)g(s) ds = m x,N Tn m x,0 ϕ x,h (s)g(s) ds = N Tn i=1 m x,i m x,i-1 ϕ x,h (s)g(s) ds,
nd writeD

E(ĝ GM n (x)) = E(ĝ GM n (x)) -l h (x) + l h (x) ∆ = ∆ x,h + l h (x). @IFTA PP Chapter 1 e (rst ontrol ∆ x,h F e hveD ∆ x,h = N Tn i=1 m x,i m x,i-1 (g(t x,i ) -g(s))ϕ x,h (s) ds. @IFUA ell tht g nd ϕ x,h re in C 2 D then ylor expnsions of g nd ϕ x,h for s in ]m x,i-1 , m x,i [ round t x,i yieldD g(s) = g(t x,i ) + (s -t x,i )g (t x,i ) + 1 2 (s -t x,i ) 2 g (θ x,i ), ndD ϕ x,h (s) = ϕ x,h (t x,i ) + (s -t x,i )ϕ x,h (t x,i ) + 1 2 (s -t x,i ) 2 ϕ x,h (s x,i ),
for some θ x,i nd s x,i etween t x,i nd sF snjeting these expnsions in @IFUA givesD

∆ x,h = N Tn i=1 g (t x,i )ϕ x,h (t x,i ) mx,i mx,i-1 (t x,i -s) ds - 1 2 N Tn i=1 ϕ x,h (t x,i ) mx,i mx,i-1 (s -t x,i ) 2 g (θ x,i ) ds - N Tn i=1 ϕ x,h (t x,i )g (t x,i ) mx,i mx,i-1 (s -t x,i ) 2 ds - 1 2 N Tn i=1 ϕ x,h (t x,i ) mx,i mx,i-1 (s -t x,i ) 3 g (θ x,i ) ds. - 1 2 N Tn i=1 g (t x,i ) mx,i mx,i-1 (s -t x,i ) 3 ϕ x,h (s x,i ) ds - 1 4 N Tn i=1 mx,i mx,i-1 (s -t x,i ) 4 g (θ x,i )ϕ x,h (s x,i ) ds. (1.8)
o ontrol these termsD we shll use the following lemm nnouned without proof @the proof is similr to tht of vemm I in fenelmdni et alF @PHIWAAF Lemma 1.4.1 Let T n = {t i,n , i = 1, • • • , n} for n ≥ 1 be a regular sequence of designs (see Denition 1.1.1) and let M n = {m i , 0 ≤ i ≤ n} (m i are the midpoints dened by @IFQA). Suppose that

M n ∩ [x -h, x + h] = ∅. If nh ≥ 1 then, sup 0≤j≤n (t j+1,n -t j,n ) = O 1 n
and N Tn = O(nh).

@IFWA ell tht g nd g re oth ounded nd tht for some pproprite onstnts c j for j = 0, 1, 2D

sup 0≤t≤1 |ϕ (j) x,h (t)| ≤ c j h j+1 . @IFIHA 1.4. PROOFS

PQ

yn the one hndD we get using @IFWA nd @IFIHAD

N Tn i=1 ϕ x,h (t x,i ) m x,i m x,i-1 (s -t x,i ) 2 g (θ x,i ) ds = O 1 n 2 . @IFIIA N Tn i=1 ϕ x,h (t x,i )g (t x,i ) m x,i m x,i-1 (s -t x,i ) 2 ds = O 1 n 2 h . @IFIPA N Tn i=1 ϕ x,h (t x,i ) m x,i m x,i-1 (s -t x,i ) 3 g (θ x,i ) ds = O 1 n 3 h . @IFIQA N Tn i=1 g (t x,i ) m x,i m x,i-1 (s -t x,i ) 3 ϕ x,h (s x,i ) ds = O 1 n 3 h 2 . @IFIRA N Tn i=1 m x,i m x,i-1 (s -t x,i ) 4 g (θ x,i )ϕ x,h (s x,i ) ds = O 1 n 4 h 2 .
@IFISA yn the other hndD the de(nition of the midpoints

(m i ) 0≤i≤n yieldsD (m x,i-1 -t x,i ) = 1 2 (t x,i-1 -t x,i ) ∆ = - 1 2 d x,i-1 for i = 1, • • • , N Tn . @IFITA
sing iqutions @IFIIAE@IFITA nd the ft tht lim n→∞ nh = ∞ we otinD

∆ x,h = 1 8 N Tn i=1 g (t x,i )ϕ x,h (t x,i )(d 2 x,i-1 -d 2 x,i ) + O 1 n 2 h . @IFIUA
o ontrol the (rst term of the right side of @IFIUAD we shll use the following lemm @its proof is given elowAF Lemma 1.4.2 Let {T n , n ≥ 1} be a regular sequence of designs generated by a density function

f (see Denition 1.1.1). If f ∈ C 2 ([0, 1]) then for i = 2, • • • , n -1, d i-1 -d i = f (t i ) 2n 2 f (t i ) 1 f 2 (t * i ) + 1 f 2 (t * i-1 ) + O 1 n 3 , @IFIVA for some t * i ∈]t i , t i+1 [, recall that d i = t i+1 -t i .
sing the previous lemm we hveD

∆ x,h = 1 16n 2 N Tn i=1 g (t x,i )ϕ x,h (t x,i ) f (t x,i ) f (t x,i ) 1 f 2 (t * x,i ) + 1 f 2 (t * x,i-1 ) (d x,i-1 + d x,i ) + O 1 n 3 + 1 n 2 h .
ell tht g , f nd 1 f re ll oundedF e otin using vemm IFRFID inequlity @IFIHAD

1 16n 2 N Tn i=1 g (t x,i )ϕ x,h (t x,i ) f (t x,i ) f (t x,i ) 1 f 2 (t * x,i ) + 1 f 2 (t * x,i-1 ) (d x,i-1 + d x,i ) = O 1 n 2 . ine lim n→∞ nh = ∞ thenD ∆ x,h = O 1 n 2 h . @IFIWA
he ontrol of l h (x) is lssil nd it n e seen from qsser nd wüller @IWVRA thtD

l h (x) = g(x) + 1 2 h 2 g (x) 1 -1 t 2 K(t) dt + o(h 2 ). @IFPHA
pinllyD olleting @IFTAD @IFIWA nd @IFPHA we otinD

Bias(ĝ GM n (x)) = 1 2 h 2 g (x)B + o(h 2 ) + O 1 n 2 h ,
where B = 1 -1 t 2 K(t) dtF his onludes the proof of roposition IFQFIF Proof of Lemma 1.4.2.

yn the one hndD we hve from the de(nition of the regulr sequene of designsD for k = 1,

• • • , n- 1D F (t k+1 ) -F (t k ) = t k+1 t k f (t) dt = 1 n .
he men vlue theorem yields tht for k

= 1, • • • , n -1D d k = t k+1 -t k = 1 nf (t * k )
, @IFPIA for some t * k ∈]t k , t k+1 [F yn the other hndD note tht for k = 2, . . . , n -1 we hveD

t k+1 t k f (t) dt - t k t k-1 f (t) dt = 1 n - 1 n = 0. ine f ∈ C 2 ([0, 1]) then ylor expnsion of f round t k yieldsD f (t k )d k + f (t k ) t k+1 t k (t -t k ) dt + 1 2 t k+1 t k (t -t k ) 2 f ( k ) dt -f (t k )d k-1 -f (t k ) t k t k-1 (t -t k ) dt - 1 2 t k t k-1 (t -t k ) 2 f (η k ) = 0, for some k ∈]t k , t k+1 [ nd some η k ∈]t k-1 , t k [F husD f (t k )(d k -d k-1 ) + 1 2 f (t k )(d 2 k + d 2 k-1 ) + 1 2 t k+1 t k (t -t k ) 2 f ( k ) dt - 1 2 t k t k-1 (t -t k ) 2 f (η k ) = 0.
pinllyD using iqution @IFPIAD vemm IFRFI nd the ft tht f is ounded we otinD

d k -d k-1 = - f (t k ) 2n 2 f (t k ) 1 f 2 (t * k ) + 1 f 2 (t * k-1 ) + O 1 n 3 .
his onludes the proof of vemm IFRFPF 1.4. PROOFS

PS

Proof of Proposition 1.3.2.

por i, j = 1, • • • , n we hveD Cov (Y (t i ), Y (t j )) = 1 m 2 m k=1 m l=1 Cov (ε k (t i ), ε l (t j )) = 1 m 2 m k=1 Cov (ε k (t i ), ε k (t j )) = 1 m R(t i , t j ).
e otin using this lst equlityD

Var ĝGM n (x) = 1 m n i=1 n j=1 R(t i , t j ) m i m i-1 ϕ x,h (t) dt m j m j-1 ϕ x,h (s) ds. ine ϕ x,h is of support [x -h, x + h]
we getD y tking the nottion @IFSA s in the proof of roposition IFQFID

Var ĝGM n (x) = 1 m N Tn i=1 N Tn j=1 R(t x,i , t x,j ) m x,i m x,i-1 ϕ x,h (t) dt m x,j m x,j-1 ϕ x,h (s) ds. vetD σ 2 x,h ∆ = x+h x-h x+h x-h ϕ x,h (s)R(t, s)ϕ x,h (t)ds dt = m x,N Tn m x,0 m x,N Tn m x,0 ϕ x,h (s)R(t, s)ϕ x,h (t)ds dt = N Tn i=1 N Tn j=1 m x,i m x,i-1 m x,j m x,j-1 ϕ x,h (s)R(t, s)ϕ x,h (t)ds dt, nd writeD Var ĝGM n (x) = Var ĝGM n (x) - 1 m σ 2 x,h + 1 m σ 2 x,h ∆ = ∆ x,h + 1 m σ 2 x,h . @IFPPA e (rst ontrol ∆ x,h F e hveD ∆ x,h = 1 m N Tn i=1 N Tn j=1 m x,i m x,i-1 m x,j m x,j-1 ϕ x,h (t)ϕ x,h (s) R(t x,i , t x,j ) -R(t, s) ds dt.
por i, j = 1, . . . , N Tn setD

I i,j = m x,i m x,i-1 m x,j m x,j-1 ϕ x,h (t)ϕ x,h (s) R(t x,i , t x,j ) -R(t, s) ds dt. @IFPQA
ine R is symmetri funtion then ∆ x,h n e written s followsF

∆ x,h = 1 m N Tn i=1 I i,i + 2 N Tn i=1 N Tn j=i+1 I i,j .
@IFPRA e (rst ontrol the digonl terms I i,i for i = 1, • • • , N Tn F ine R is symmetri funtionD it su0es to onsider the integrl over the lower hlf @tringulrA of the squre [m x,i-1 , m x,i ] 2 F his tringulr is further split into three tringulrs s followsD

D i,1 = {(t, s) : m x,i-1 ≤ s ≤ t < t x,i }. D i,2 = {(t, s) : m x,i-1 ≤ s < t x,i < t < m x,i }. D i,3 = {(t, s) : t x,i < s ≤ t < m x,i }. t s m x,i m x,i-1 m x,i t x,i m x,i-1 D i,3 D i,1 D i,2 t x,i he term I i,i is then writtenD I i,i = 2 3 k=1 D i,k ϕ x,h (t)ϕ x,h (s) R(t x,i , t x,i ) -R(t, s) ds dt ∆ = 2 3 k=1 I (k) i,i .
e onsider (rst the term I

i,i F e hveD

I (1) i,i = t x,i m x,i-1 t m x,i-1 ϕ x,h (t)ϕ x,h (s) R(t x,i , t x,i ) -R(t, s) ds dt. @IFPSA feuse of essumptions (B) nd (C)D we n expnd R in ylor series round (t x,i , t x,i ) for (t, s) in D i,1 s followsD R(t, s) = R(t x,i , s) + (t -t x,i )R (1,0) (t x,i , s) + 1 2 (t -t x,i ) 2 R (2,0) ( (1)
x,i , s)

= R(t x,i , t x,i ) + (s -t x,i )R (0,1) (t x,i , t - x,i ) + 1 2 (s -t x,i ) 2 R (0,2) (t x,i , η (1)
x,i )

+ (t -t x,i )R (1,0) (t x,i , t - x,i ) + (t -t x,i )(s -t x,i )R (1,1) (t x,i , η (2)
x,i )

+ 1 2 (t -t x,i ) 2 R (2,0) ( (1)
x,i , s),

for some

(1)

x,i in ]t, t x,i [ nd some η

(1)

x,i , η

x,i in ]s, t x,i [F e otin y inserting the ove eqution in @IFPSAD

I (1) i,i = R (0,1) (t x,i , t - x,i ) t x,i m x,i-1 t m x,i-1 ϕ x,h (t)ϕ x,h (s)(t x,i -s) ds dt + R (1,0) (t x,i , t - x,i ) t x,i m x,i-1 t m x,i-1 ϕ x,h (t)ϕ x,h (s)(t x,i -t) ds dt - 1 2 t x,i m x,i-1 t m x,i-1 ϕ x,h (t)ϕ x,h (s) (s -t x,i ) 2 R (0,2) (t x,i , η (1)
x,i )

+ 2(t -t x,i )(s -t x,i )R (1,1) (t x,i , η (2)
x,i ) + (t -t x,i ) 2 R (2,0) ( (1)
x,i , s) ds dt

∆ = I (1,1) i,i + I (1,2) i,i + I (1,3) i,i . @IFPTA ine ϕ x,h is in C 2 then ylor expnsion of ϕ x,h round t x,i for t nd s in D i,1 givesD ϕ x,h (t) = ϕ x,h (t x,i ) + (t -t x,i )ϕ x,h (t x,i ) + 1 2 (t -t x,i ) 2 ϕ x,h ( (2)
x,i ),

ndD ϕ x,h (s) = ϕ x,h (t x,i ) + (s -t x,i )ϕ x,h (t x,i ) + 1 2 (s -t x,i ) 2 ϕ x,h (η (3)
x,i ),

1.4. PROOFS PU for some

(2)

x,i in ]t, t x,i [ nd some η

(3)

x,i in ]s, t x,i [F snserting these expnsions in I

(1,1) i,i ove yieldsD

I (1,1) i,i = R (0,1) (t x,i , t - x,i )ϕ 2 x,h (t x,i ) t x,i m x,i-1 t m x,i-1 (t x,i -s) ds dt -R (0,1) (t x,i , t - x,i )ϕ x,h (t x,i )ϕ x,h (t x,i ) t x,i m x,i-1 t m x,i-1 (t x,i -s) 2 ds dt + R (0,1) (t x,i , t - x,i )ϕ x,h (t x,i )ϕ x,h (t x,i ) t x,i m x,i-1 t m x,i-1 (t -t x,i )(t x,i -s) ds dt + 1 2 R (0,1) (t x,i , t - x,i )ϕ x,h (t x,i ) t x,i m x,i-1 t m x,i-1 ϕ x,h (η (3) x,i )(t x,i -s) 3 ds dt -R (0,1) (t x,i , t - x,i )ϕ x,h 2 (t x,i ) t x,i m x,i-1 t m x,i-1 (t -t x,i )(t x,i -s) 2 ds dt + 1 2 R (0,1) (t x,i , t - x,i )ϕ x,h (t x,i ) t x,i m x,i-1 t m x,i-1 ϕ x,h (η (3) x,i )(t -t x,i )(t x,i -s) 3 ds dt + 1 2 R (0,1) (t x,i , t - x,i )ϕ x,h (t x,i ) t x,i m x,i-1 t m x,i-1 ϕ x,h ( (2)
x,i )(t -t x,i ) 2 (t x,i -s) ds dt - 1 2 R (0,1) (t x,i , t - x,i )ϕ x,h (t x,i ) t x,i m x,i-1 t m x,i-1 ϕ x,h ( (2)
x,i )(t -t x,i ) 2 (t x,i -s) 2 ds dt + 1 4 R (0,1) (t x,i , t - x,i ) t x,i m x,i-1 t m x,i-1 ϕ x,h (η (2) x,i )ϕ x,h ( (2)
x,i )(t -t x,i ) 2 (t x,i -s) 3 ds dt.
e otin using essumption (C)D vemm IFRFID snequlity @IFIHA nd the ft tht lim

n→∞ nh = ∞D I (1,1) i,i = R (0,1) (t x,i , t - x,i )ϕ 2 x,h (t x,i ) t x,i m x,i-1 t m x,i-1 (t x,i -s) ds dt + O 1 n 4 h 3 .
st is esy to verify tht for l, l ∈ {0, 1, 2} we hveD

tx,i mx,i-1 t mx,i-1 (t -t x,i ) l (s -t x,i ) l ds dt = (m x,i-1 -t x,i ) l+l +2 1 (l + 1)(l + 1) - 1 (l + l + 2)(l + 1)
.

From Equation (1.16) we get,

tx,i mx,i-1 t mx,i-1 (t -t x,i ) l (s -t x,i ) l ds dt = -d x,i-1 2 l+l +2 1 (l + 1)(l + 1) - 1 (l + l + 2)(l + 1)
.

(1.27) e otin using iqution @IFPUAD

I (1,1) i,i = 1 24 R (0,1) (t x,i , t - x,i )ϕ 2 x,h (t x,i )d 3 x,i-1 + O 1 n 4 h 3 iqution @IFPIA then yieldsD for some t * x,i-1 in ]t x,i-1 , t x,i [D I (1,1) i,i = 1 24n 2 ϕ 2 x,h (t x,i ) f 2 (t * x,i-1 ) R (0,1) (t x,i , t - x,i )d x,i-1 + O 1 n 4 h 3 ,

PV

Chapter 1 e otin using vemm IFRFID

N Tn i=1 I (1,1) i,i = 1 24n 2 N Tn i=1 ϕ 2 x,h (t x,i ) f 2 (t * x,i-1 ) R (0,1) (t x,i , t - x,i )d x,i-1 + O 1 n 3 h 2 .
sing lssil pproximtion of sum y n integrl @see for instneD fenelmdni et al.

@PHIWA FfF vemm P thereA we otinD

N Tn i=1 I (1,1) i,i = 1 24n 2 x+h x-h R (0,1) (t, t -) ϕ 2 x,h (t) f 2 (t) dt + O 1 n 3 h 2 .
@IFPVA sn similr wyD we verify thtD

N Tn i=1 I (1,2) i,i = 1 48n 2 x+h x-h R (1,0) (t, t -) ϕ 2 x,h (t) f 2 (t) dt + O 1 n 3 h 2 . @IFPWA essumption (C)D vemm IFRFI nd snequlity @IFIHA yieldD N Tn i=1 I (1,3) i,i = O 1 n 3 h . @IFQHA
e otin olleting @IFPVAD @IFPVAD @IFPWA nd @IFQHAD

N Tn i=1 I (1) i,i = 1 24n 2 x+h x-h R (0,1) (t, t -) + 1 2 R (1,0) (t, t -) ϕ 2 x,h (t) f 2 (t) dt + O 1 n 3 h 2 .
@IFQIA imilrly we otinD

N Tn i=1 I (2) i,i = 1 16n 2 x+h x-h R (0,1) (t, t -) -R (1,0) (t, t -) ϕ 2 x,h (t) f 2 (t) dt + O 1 n 3 h 2 , @IFQPA ndD N Tn i=1 I (3) i,i = - 1 24n 2 x+h x-h 1 2 R (0,1) (t + , t) + R (1,0) (t + , t) ϕ 2 x,h (t) f 2 (t) dt + O 1 n 3 h 2 .
@IFQQA pinllyD summing @IFQIAD @IFQPA nd @IFQQA nd using essumption (B) we otinD

N Tn i=1 I i,i = 2 N Tn i=1 3 k=1 I (k) i,i = 1 24n 2 x+h x-h ϕ 2 x,h (t) f 2 (t) 5R (0,1) (t, t -) -2R (1,0) (t, t -) -R (0,1) (t + , t) -2R (1,0) (t + , t) dt + O 1 n 3 h 2 = 1 6n 2 x+h x-h ϕ 2 x,h (t) f 2 (t) R (1,0) (t -, t) -R (1,0) (t + , t) dt + O 1 n 3 h 2 = 1 6n 2 x+h x-h ϕ 2 x,h (t) f 2 (t) α(t) dt + O 1 n 3 h 2 . @IFQRA 1.4. PROOFS PW e now ontrol the o' digonl terms I i,j given y @IFPQA for 1 ≤ i < j ≤ N Tn F sing essumption (C)D ylorEexpnsion of R t (t x,i , t x,j ) for (t, s) ∈ [m x,i-1 , m x,i ] × [m x,j-1 , m x,j ] givesD R(t, s) = R(t x,i , t x,j ) + (t -t x,i )R (1,0) (t x,i , t x,j ) + (s -t x,j )R (0,1) (t x,i , t x,j ) + 1 2 (t -t x,i ) 2 R (2,0) ( x,i , t x,j ) + 1 2 (s -t x,j ) 2 R (0,2) (t x,i , η x,j ) + (t -t x,i )(s -t x,j )R (1,1) ( x,i , η x,j ),
for some x,i etween t nd t x,i nd some η x,j etween s nd t x,j F etD

I i,j = I (1) i,j + I (2) i,j + I (3) i,j + I (4) i,j + I (5) i,j , @IFQSA whereD I (1) i,j = R (1,0) (t x,i , t x,j ) m x,i m x,i-1 m x,j m x,j-1 (t x,i -t)ϕ x,h (t)ϕ x,h (s) ds dt. @IFQTA I (2) i,j = R (0,1) (t x,i , t x,j ) m x,i m x,i-1 m x,j m x,j-1 (t x,j -s)ϕ x,h (t)ϕ x,h (s) ds dt. I (3) i,j = - 1 2 m x,i m x,i-1 m x,j m x,j-1 ϕ x,h (t)ϕ x,h (s)(t -t x,i ) 2 R (2,0) ( x,i , t x,j ) ds dt. I (4) i,j = - 1 2 m x,i m x,i-1 m x,j m x,j-1 ϕ x,h (t)ϕ x,h (s)(s -t x,j ) 2 R (0,2) (t x,i , η x,j ) ds dt. I (5) i,j = - m x,i m x,i-1 m x,j m x,j-1 ϕ x,h (t)ϕ x,h (s)(t -t x,i )(s -t x,j )R (1,1) ( x,i , η x,j ) ds dt.
e (rst ontrol the term

I (1) i,j for i, j = 1, • • • , N Tn F ine ϕ x,h is in C 2 D ylor expnsions of ϕ x,h round t x,i for t ∈]m x,i-1 , m x,i [ yieldsD ϕ x,h (t) = ϕ x,h (t x,i ) + (t -t x,i )ϕ x,h (t x,i ) + 1 2 (t -t x,i ) 2 ϕ x,h ( (1)
x,i ), @IFQUA for some

(1)

x,i etween t x,i nd tF imilrlyD for s ∈]m x,j-1 , m x,j [ we otinD

ϕ x,h (s) = ϕ x,h (t x,j ) + (s -t x,j )ϕ x,h (t x,j ) + 1 2 (s -t x,j ) 2 ϕ x,h (η (1)
x,j ), @IFQVA for some η

(1)

x,j etween t x,j nd sF sing @IFQTAD @IFQUA nd @IFQVA we otinD

I (1) i,j = R (1,0) (t x,i , t x,j )ϕ x,h (t x,i )ϕ x,h (t x,j ) m x,i m x,i-1 m x,j m x,j-1 (t x,i -t) ds dt + R (1,0) (t x,i , t x,j )ϕ x,h (t x,i )ϕ x,h (t x,j ) m x,i m x,i-1 m x,j m x,j-1 (t x,i -t)(s -t x,j ) ds dt -R (1,0) (t x,i , t x,j )ϕ x,h (t x,i )ϕ x,h (t x,j ) m x,i m x,i-1 m x,j m x,j-1 (t -t x,i ) 2 ds dt + 1 2 R (1,0) (t x,i , t x,j )ϕ x,h (t x,i ) m x,i m x,i-1 m x,j m x,j-1 (t x,i -t)(s -t x,j ) 2 ϕ x,h (η (1) x,j ) ds dt -R (1,0) (t x,i , t x,j )ϕ x,h (t x,i )ϕ x,h (t x,j ) m x,i m x,i-1 m x,j m x,j-1 (t -t x,i ) 2 (s -t x,j ) ds dt - 1 2 R (1,0) (t x,i , t x,j )ϕ x,h (t x,i ) m x,i m x,i-1 m x,j m x,j-1 (t -t x,i ) 2 (s -t x,j ) 2 ϕ x,h (η (1)
x,j ) ds dt

- 1 2 R (1,0) (t x,i , t x,j )ϕ x,h (t x,j ) m x,i m x,i-1 m x,j m x,j-1 (t -t x,i ) 3 ϕ x,h ( (1)
x,i ) ds dt

- 1 2 R (1,0) (t x,i , t x,j )ϕ x,h (t x,j ) m x,i m x,i-1 m x,j m x,j-1 (t -t x,i ) 3 (s -t x,j )ϕ x,h ( (1)
x,i ) ds dt

- 1 4 R (1,0) (t x,i , t x,j ) m x,i m x,i-1 m x,j m x,j-1 (t -t x,i ) 3 (s -t x,j ) 2 ϕ x,h ( (1)
x,i )ϕ x,h (η

x,j ) ds dt.

e otin using essumption (C)D vemm IFRFI nd snequlity @IFIHAD

I (1) i,j = R (1,0) (t x,i , t x,j )ϕ x,h (t x,i )ϕ x,h (t x,j ) m x,i m x,i-1 m x,j m x,j-1 (t x,i -t) ds dt + R (1,0) (t x,i , t x,j )ϕ x,h (t x,i )ϕ x,h (t x,j ) m x,i m x,i-1 m x,j m x,j-1 (t x,i -t)(s -t x,j ) ds dt -R (1,0) (t x,i , t x,j )ϕ x,h (t x,i )ϕ x,h (t x,j ) m x,i m x,i-1 m x,j m x,j-1 (t -t x,i ) 2 ds dt + O 1 n 5 h 4 ∆ = I (1,1) i,j + I (1,2) i,j + I (1,3) i,j + O 1 n 5 h 4 .
@IFQWA e (rst ontrol the term I

(1,1) i,j F fsi integrtion together with iqution @IFITA yieldD

I (1,1) i,j = - 1 8 R (1,0) (t x,i , t x,j )ϕ x,h (t x,i )ϕ x,h (t x,j )(m x,j -m x,j-1 )(d 2 x,i -d 2 x,i-1 ).
sing the de(nition of the midpoints (m x,j ) j nd vemm IFRFP we otinD for some

t * x,i in ]t x,i , t x,i+1 [D d 2 x,i -d 2 x,i-1 = (d x,i + d x,i-1 )(d x,i -d x,i-1 ) = 2(m x,i -m x,i-1 )(d x,i -d x,i-1 ) = -(m x,i -m x,i-1 ) f (t x,i ) n 2 f (t x,i ) 1 f 2 (t * x,i ) + 1 f 2 (t * x,i-1 ) + O 1 n 4 . @IFRHA = O 1 n 3 @IFRIA 1.4. PROOFS

QI

sing iqution @IFRIAD snequlity @IFIHAD essumption (C)D vemm IFRFP nd the ft tht f, f nd 1 f re ll oundedD

I (1,1) i,j = O 1 n 4 h 2 . vemm IFRFI then yieldsD N Tn i=1 N Tn j=i+1 I (1,1) i,j = O 1 n 2 .
@IFRPA e ontrol now the term I

(1,2) i,j . fsi integrtion together with iqution @IFITA yieldD

I (1,2) i,j = - 1 64 R (1,0) (t x,i , t x,j )ϕ x,h (t x,i )ϕ x,h (t x,j )(d 2 x,i -d 2 x,i-1 )(d 2 x,j -d 2 x,j-1 ).
sing iqutions @IFRHA nd @IFRIAD snequlity @IFIHAD essumption (C)D vemm IFRFP nd the oundedness of f, f nd 1 f we getD

I (1,2) i,j = - 1 64n 4 R (1,0) (t x,i , t x,j )ϕ x,h (t x,i )ϕ x,h (t x,j )(m x,i -m x,i-1 )(m x,j -m x,j-1 ) × f (t x,i ) f (t x,i ) 1 f 2 (t * x,i ) + 1 f 2 (t * x,i-1 ) f (t x,j ) f (t x,j ) 1 f 2 (t * x,j ) + 1 f 2 (t * x,j-1 ) + O 1 n 7 h 3 = O 1 n 6 h 3 . husD N Tn i=1 N Tn j=i+1 I (1,2) i,j = O 1 n 4 h . @IFRQA
e ontrol now the term I

(1,3) i,j F sing iqution @IFITA yieldsD

I (1,3) i,j = - 1 24 R (1,0) (t x,i , t x,j )ϕ x,h (t x,i )ϕ x,h (t x,j )(m x,j -m x,j-1 )(d 3 x,i + d 3 x,i-1 ).
@IFRRA e otin using the de(nition of the midpoints together with iqution @IFPIAD

d 3 x,i + d 3 x,i-1 = (d x,i + d x,i-1 )(d 2 x,i -d x,i d x,i-1 + d 2 x,i-1 ) = 2 n 2 (m x,i -m x,i-1 ) 1 f 2 (t * x,i ) - 1 f (t * x,i )f (t * x,i-1 ) + 1 f 2 (t * x,i-1 )
,

for some t * x,i in ]t x,i , t x,i+1 [F husD N Tn i=1 N Tn j=i+1 I (1,3) i,j = - 1 12n 2 N Tn i=1 N Tn j=1 i<j R (1,0) (t x,i , t x,j )ϕ x,h (t x,i )ϕ x,h (t x,j )(m x,j -m x,j-1 ) × (m x,i -m x,i-1 ) 1 f 2 (t * x,i ) - 1 f (t * x,i )f (t * x,i-1 ) + 1 f 2 (t * x,i-1 )
.

sing @twieA the lssil pproximtion of sum y n integrl @see for instne vemm I in fenelmdni et al. @PHIWA we otinD

N Tn i=1 N Tn j=i+1 I (1,3) i,j = - 1 12n 2 x+h x-h s x-h R (1,0) (t, s)ϕ x,h (s) ϕ x,h (t) f 2 (t) dt ds + O 1 n 3 h 2 .
(1.45) e otin olleting @IFQWAD @IFRPAD @IFRQA nd @IFRSAD

N Tn i=1 N Tn j=i+1 I (1) i,j = - 1 12n 2 x+h x-h s x-h R (1,0) (t, s)ϕ x,h (s) ϕ x,h (t) f 2 (t) dt ds + O 1 n 3 h 2 + 1 n 2 . @IFRTA e shll prove thtD 1 12n 2 x+h x-h s x-h R (1,0) (t, s)ϕ x,h (s) ϕ x,h (t) f 2 (t) dt ds = - 1 12n 2 x+h x-h R (1,0) (s -, s)ϕ 2 x,h (s) 1 f 2 (s)
ds.

@IFRUA por thisD we use prtil integrl to get for

s ∈]x -h, x + h[D s x-h R (1,0) (t, s) ϕ x,h (t) f 2 (t) dt = R (1,0) (t, s) ϕ x,h (t) f 2 (t) s x-h - s x-h ϕ x,h (t) 1 f 2 (t) R (2,0) (t, s) - 2f (t) f 3 (t) R (1,0) (t, s) dt.
ell tht ϕ x,h (x -h) = 0F e otin using snequlity @IFIHAD essumptions @gA nd the ft tht f nd 1 f re oth oundedD

s x-h ϕ x,h (t) 1 f 2 (t) R (2,0) (t, s) - 2f (t) f 3 (t) R (1,0) (t, s) dt = O(1). husD s x-h R (1,0) (t, s) ϕ x,h (t) f 2 (t) dt = R (1,0) (s -, s) ϕ x,h (s) f 2 (s) + O(1)
. @IFRVA pinllyD using @IFRTA nd @IFRVA we otinD

N Tn i=1 N Tn j=i+1 I (1) i,j = - 1 12n 2 x+h x-h R (1,0) (s -, s) ϕ 2 x,h (s) f 2 (s) ds + O 1 n 3 h 2 + 1 n 2 .
@IFRWA imilrly we prove thtD

N Tn i=1 N Tn j=i+1 I (2) i,j = - 1 12n 2 x+h x-h x+h t R (0,1) (t, s)ϕ x,h (t)ϕ x,h (s) 1 f 2 (s) ds dt + O 1 n 3 h 2 + 1 n 2 = 1 12n 2 x+h x-h R (0,1) (t, t + ) ϕ 2 x,h (t) f 2 (t) dt + O 1 n 3 h 2 + 1 n 2 .
@IFSHA 1.4. PROOFS QQ e verify using essumption (C)D vemm IFRFI nd snequlity @IFIHA thtD

N Tn i=1 N Tn j=i+1 I (3) i,j = O 1 n 2 , N Tn i=1 N Tn j=1 i<j I (4) i,j = O 1 n 2 nd N Tn i=1 N Tn j=1 i<j I (5) i,j = O 1 n 2 .
@IFSIA sing @IFQSAD @IFRWAD @IFSHA nd @IFSIA yieldsD

2 N Tn i=1 N Tn j=i+1 I i,j = - 1 6n 2 x+h x-h R (1,0) (t -, t) -R (0,1) (t, t + ) ϕ 2 x,h (t) f 2 (t) dt + O 1 n 2 . @IFSPA ine R (1,0) (t -, t) = lim s↑t ∂R(s,t) ∂s = lim s↑t ∂R(t,s) ∂s = R (0,1) (t, t-) thenD 2 N Tn i=1 N Tn j=i+1 I i,j = - 1 6n 2 x+h x-h α(t) ϕ 2 x,h (t) f 2 (t) dt + O 1 n 2 + 1 n 3 h 2 .
@IFSQA pinllyD we otin y @IFPRAD @IFQRA nd @IFSQAD

∆ x,h = O 1 mn 2 + 1 mn 3 h 2 . @IFSRA por the ontrol of 1 m σ 2
x,h D we use its symptoti expression given y fenhenni nd hdi @PHHUA yD

σ 2 x,h = R(x, x) - 1 2 α(x)C K h + o(h). @IFSSA
pinllyD the proof of roposition IFQFP is onluded y olleting @IFPPAD @IFSRA nd @IFSSAF Proof of Proposition 1.3.3.

vet

I 1 = 1 0 R(x, x)w(x) dx nd putD Ψ(h, m) = - C K h 2m 1 0 α(x)w(x) dx + 1 4 h 4 B 2 1 0 [g (x)] 2 w(x) dx.
e hve from heorem IFQFID

IMSE(h) = I 1 m + Ψ(h, m) + o h 4 + h m + O h n 2 + 1 n 4 h 2 + 1 mn 3 h 2 + 1 mn 2 . vet h * e s de(ned in @IFRAF st is ler tht h * = argmin 0<h<1 Ψ(h, m) so tht Ψ(h, m) ≥ Ψ(h * , m) for every 0 < h < 1F vet h n,m e s de(ned in gorollry IFQFQF e hveD IMSE(h * ) IMSE(h n,m ) = I1 m + Ψ(h * , m) + o h * 4 + h * m + O h * n 2 + 1 n 4 h * 2 + 1 mn 3 h * 2 + 1 mn 2 I1 m + Ψ(h n,m , m) + o h 4 n,m + hn,m m + O hn,m n 2 + 1 n 4 h 2 n,m + 1 mn 3 h 2 n,m + 1 mn 2 ≤ I 1 + mΨ(h * , m) + o mh * 4 + h * + O mh * n 2 + m n 4 h * 2 + 1 n 3 h * 2 + 1 n 2 I 1 + mΨ(h n,m , m) + o mh 4 n,m + h n,m + O mhn,m n 2 + m n 4 h 2 n,m + 1 n 3 h 2 n,m + 1 n 2 . xote tht mΨ(h n,m , m) = O(h n,m )F sing the de(nition of h * , the fts thtX mh 3 n,m = O(1)D lim n,m→∞ h n,m = 0 nd the ssumption m n = O(1) s n, m → ∞D we otinD lim n,m→∞ IMSE(h * ) IMSE(h n,m ) ≤ 1.
his onludes the proof of roposition IFQFQF Proof of Theorem 1.3.2.

vet x ∈]0, 1[ e (xedF e hveD √ m ĝGM n,m (x) -g(x) = √ m ĝGM n,m (x) -E ĝGM n,m (x) + √ m Bias ĝGM n,m (x) . (1.56) ine lim n,m→∞ √ mh 2 = 0 nd m n = O(1) s n, m → ∞ then roposition IFQFI implies thtD lim n,m→∞ √ m Bias ĝGM n,m (x) = 0. @IFSUA
gonsider now the (rst term of the right side of @IFSTAF e tke the sme nottion s in the proof of roposition IFQFI nd rell tht

ϕ x,h is of support [x -h, x + h]F ine Y (t x,i ) -E Y (t x,i ) = ε(t x,i )D we hveD s done y primn nd érez srirren @IWWIAD √ m ĝGM n,m (x) -E ĝGM n,m (x) = 1 √ m m j=1 N Tn i=1 mx,i mx,i-1 ϕ x,h (s) ds ε j (t x,i ) = 1 √ m m j=1 N Tn i=1 mx,i mx,i-1 ϕ x,h (s) ds ε j (t x,i ) -ε j (x)) + 1 √ m N Tn i=1 mx,i mx,i-1 ϕ x,h (s) ds m j=1 ε j (x).
(1.58) e strt y ontrolling the seond term of the right side of @QFTUAF yn the one hndD we otin using lssil pproximtion of sum y n integrl @see for instneD fenelmdni et al.

@PHIWA FfF vemm P thereAD

N Tn i=1 m x,i m x,i-1 ϕ x,h (s) ds -→ n→∞ 1 -1 K(t) dt = 1.
yn the other hndD the gentrl vimit heorem for iFiFdF vriles yieldsD

1 √ m m j=1 ε j (x) D -→ m→∞ Z where Z ∼ N (0, R(x, x)).
e shll prove now tht the (rst term of the right side of @QFTUA tends to H in proility s n, m tends to in(nityF vetD

A m,n (x) = 1 √ m m j=1 N Tn i=1 m x,i m x,i-1 ϕ x,h (s) ds ε j (t x,i ) -ε j (x)) ∆ = 1 √ m m j=1
T n,j (x).

prom gheyshev9s snequlityD it su0es to prove tht lim n,m→∞

E(A 2 m,n (x)) = 0F e hve for j = lD E(ε j (x)ε l (y)) = 0 so E(T n,j (x)T n,l (x)) = 0F reneD E(A 2 m,n (x)) = 1 m m j=1 m l=1 E(T n,j (x)T n,l (x)) = 1 m m j=1 E(T 2 n,j (x)). 1.4. PROOFS QS e hveD E(T 2 n,j (x)) = N Tn i=1 N Tn k=1 m x,i m x,i-1 ϕ x,h (t) dt m x,k m x,k-1 ϕ x,h (s) ds E ε j (t x,i ) -ε j (x) ε j (t x,k ) -ε j (x) = N Tn i=1 N Tn k=1 m x,i m x,i-1 ϕ x,h (t) dt m x,k m x,k-1 ϕ x,h (s) ds R(t x,i , t x,k ) -R(t x,i , x) -R(x, t x,k ) + R(x, x) .
ine E((T 2 n,j (x)) does not depend on j we getD

E(A 2 m,n (x)) = N Tn i=1 N Tn k=1 mx,i mx,i-1 ϕ x,h (t) dt m x,k m x,k-1 ϕ x,h (s) ds R(t x,i , t x,k ) -R(t x,i , x) -R(x, t x,k ) + R(x, x) ∆ = B n,1 (x) -B n,2 (x) -B n,3 (x) + B n,4 (x)
.

(1.59) e otin using lssil pproximtion of sum y n integrlD

B n,1 (x) = N Tn i=1 m x,i m x,i-1 ϕ x,h (t) dt x+h x-h ϕ x,h (t)R(t x,i , t) dt + O 1 nh = x+h x-h ϕ x,h (t) N Tn i=1 m x,i m x,i-1 ϕ x,h (t) dt R(t x,i , t) dt + O 1 nh = x+h x-h x+h x-h ϕ x,h (s)ϕ x,h (t)R(s, t) ds dt + O( 1 nh ) = σ 2 x,h + O 1 nh .
sing iqution @IFSSA we otinD

B n,1 (x) = R(x, x) - 1 2 α(x)C K h + o(h) + O 1 nh ,
where

C K = 1 -1 1 -1 |u -v|K(u)K(v)dudv. ine lim n→∞ h = 0 nd lim n→∞ nh = ∞F husD lim n→∞ B n,1 (x) = R(x, x). @IFTHA
gonsider now the term B n,2 (x)D we otin using lssil pproximtion of sum y n integrl @see for instneD fenelmdni et al. @PHIWA FfF vemm P thereA twieD

B n,2 (x) = x+h x-h x+h x-h ϕ x,h (s)ϕ x,h (t)R(s, x) ds dt + O 1 nh = x+h x-h ϕ x,h (s)R(s, x) ds + O 1 nh = 1 -1 K(s)R(x -sh, x) ds + O 1 nh = 0 -1 K(s)R(x -sh, x) ds + 1 0 K(s)R(x -sh, x) ds + O 1 nh . por ny s ∈] -1, 0[D ylor expnsion of R(•, x) round x yieldsD R(x -sh, x) = R(x, x) -shR (1,0) (x + , x) + o(h). QT Chapter 1 imilrlyD for s ∈]0, 1[ we otinD R(x -sh, x) = R(x, x) -shR (1,0) (x -, x) + o(h).
sing essumption (C) we getD

B n,2 (x) = R(x, x) -hR (1,0) (x + , x) 0 -1 s K(s) ds -hR (1,0) (x -, x) 1 0 s K(s) ds + o(h) + O 1 nh . reneD lim n→∞ B n,2 (x) = R(x, x). @IFTIA imilrlyD lim n→∞ B n,3 (x) = R(x, x). @IFTPA
st is esy to see thtD

lim n→∞ B n,4 (x) = lim n→∞ R(x, x) N Tn i=1 N Tn k=1 m x,i m x,i-1 ϕ x,h (t) dt m x,k m x,k-1 ϕ x,h (s) ds = R(x, x) 1 -1 K(t) dt 2 = R(x, x). @IFTQA
sing @IFSWAD @IFTHAD @IFTIAD @IFTPA nd @IFTQA we otinD

lim n,m→∞ E(A 2 m,n (x)) = 0.
his onludes the proof of heorem IFQFPF Chapter 2

Reproducing kernel Hilbert Space approach in nonparametric regression problems with correlated observations Abstract: sn this hpter we investigte the prolem of estimting the regression funtion in models with orrelted oservtionsF he dt is otined from severl experimentl unitsD eh of them forms time seriesF sing the properties of the eproduing uernel rilert spesD we onstrut new estimtor sed on the inverse of the utoovrine mtrix of the oservtionsF e give the symptoti expressions of its is nd its vrineF sn dditionD we give theoretil omprison etween this new estimtor nd the populr one proposed y qsser nd wüllerD we show tht the proposed estimtor hs n symptotilly smller vrine then the lssil oneF pinllyD we ondut simultion study to investigte the performne of the proposed estimtor nd to ompre it to the qsser nd wüller9s estimtor in (nite smple setF Keywords. Nonparametric regression, correlated observations, growth curve, reproducing kernel Hilbert space, projection estimator, asymptotic normality F Résumé: hns e hpitreD nous onsidérons le prolème d9estimtion de l fontion de réE gression dns un modèle ve des erreurs orréléesF ves données sont otenues à prtir de plusieurs unités expérimentlesD hune représente une série temporelleF in utilisnt les proE priétés de l9espe de rilert à noyu utoreproduisntD nous onstruisons un nouvel estimteur sé sur l9inverse de l mtrie d9utoovrine des oservtionsF xous donnons les expresE sions symptotiques de son iis et de s vrineF in plusD nous fisons une omprison théorique ve l9estimteur lssiqueD proposé pr qsser et wüllerF xous montrons que l vrine de l9estimteur proposé est symptotiquement plus petite que elle de l9estimteur lsE siqueF pinlementD nous e'etuons une étude de simultionD (n d9étudier l performne de l9estimteur proposéD et de le omprer ve l9estimteur de qsser et wüller pour di'érentes tilles d9éhntillonsF

Mots clés: Régression non paramétrique, observations corrélées, courbe de croissance, espace de Hilbert à noyau autoreproduisant, estimateur de projection, normalité asymptotiqueF 2.1 Introduction yne of the situtions tht sttistiins enounter in their studies is the estimtion of whole funtion sed on prtil oservtions of this funtionF por instneD in phrmokinetis one wishes to estimte the onentrtionEtime of some injeted mediine in the orgnismD sed on the oservtions of the onentrtion from lood tests over period of timeF sn sttistil termsD one wnts to estimte funtionD sy gD relting two rndom vrilesX the explntory vrile X nd the response vrile Y D without ny prmetri restritions on the funtion gF he sttistil model often used is the followingX

Y i = g(X i ) + ε i where (X i , Y i ) 1≤i≤n re n independent replites of (X, Y ) nd {ε i , i = 1, • • •
, n} re entered rndom vriles @lled errorsAF he most intensively treted model hs een the one in whih (ε i ) 1≤i≤n re independent errors nd (X i ) 1≤i≤n re (xed within some dominF e mention the works of riestly nd gho @IWUPAD fenedetti @IWUUA nd qsser nd wüller @IWUWA mong othersF roweverD the independene of the oservtions is not lwys relisti ssumptionF por instneD the growth urve models re usully used in the se of longitudinl dtD where the sme experimentl unit is eing oserved on multiple points of timeF es rel life exmpleD the heights oserved on the sme hild re orreltedF he temperture oservtions mesured long the dy re lso orreltedF por thisD we fousD in this hpterD on the nonprmetri kernel estimtion prolem where the oservtions re orreltedF sn the urrent hpterD we onsider sitution where the dt is generted from m experiE mentl units eh of them hving n mesurements of the responseF por this dtD we onsider the soElled (xed design regression model with repeted mesurements given yD

Y j (t i ) = g(t i ) + ε j (t i ) for i = 1, • • • , n nd j = 1, • • • , m, @PFIA
where {ε j , j = 1, • • • , m} is sequene of iFiFdF entered error proesses with the sme distriution s proess εF he non orreltion of the errors {ε j , j = 1, • • • , m} is nturl ssumption sine it is equivlent to ssuming tht the experimentl units @in generl individulsA re independentF his model is usully used in the growth urve nlysis nd dose response prolemsD see for instneD the work of ezzlini @IWVRAF st hs lso een onsidered y wüller @IWVRA with m = 1D where he supposed tht the oservtions re symptotilly unorrelted when the numer of oservtions tends to in(nityD iFeFD Cov(ε(s), ε(t)) = O(1/n) for s = tD whih is not relisti ssumptionD for instneD in the growth urve nlysis nd tempertureF he orrelted oservtions se ws onsidered y rrt nd herly @IWVTAD who investigted the estimtion of g in wodel @PFIA where ε is sttionry error proessF sing the kernel estimtor proposed y qsser nd wüllerD see qsser nd wüller @IWUWAD they proved the onsisteny in L 2 spe of this estimtorD when the numer of experimentl units m tends to in(nityD ut not when n tends to in(nity s in the se of independent oservtionsF he ssumption of sttionrity mde on the oservtions is however restritiveF sn the previE ous phrmokinetis exmple for instneD it is ler tht the onentrtion of the mediine will e high t the eginning then dereses with timeF por thisD we shll investigte the estimtion of g in wodel @PFIA where ε is not neessrily sttionry error proessF his se ws prtilly investigted y fenhenni nd hdi @PHITA nd perreir et al. @IWWUAD where the qsser nd wüller9s estimtor ws usedF sn this hpterD we propose new estimtor for the regression funtion g in wodel @PFIAF his estimtorD whih is lso liner kernel estimtorD is sed on the inverse of the utoovrine mtrix of the oservtionsD tht we ssume known nd invertileF 2.1. INTRODUCTION QW he proposed estimtor ws inspired y the work of ks nd lvisker @IWTTD IWVTVD IWUHA ut in di'erent ontext thn oursF hey onsidered the prmetri modelX Y (t) = βf (t) + ε(t) where β is n unknown rel prmeter nd f is known funtion elonging to the eproduing uernel rilert pe ssoited to the utoovrine funtion of the error proess εD denoted y ur@RAF hey lso ssumed tht the utoovrine mtrix is known nd invertileF st is worth noting tht the eproduing uernel rilert pes hve een used in severl dominsD for instneD in ttistis y ks nd lvisker @IWTTA nd more reently y hette et al. @PHITAD in wthemtil enlysis in hwrtz @IWTRA nd in ignl roessing in msy nd ilvermn @PHHSAF e lso give the symptoti sttistil performne of the proposed estimtor nd we ompre it to the lssil qsser nd wüller9s estimtor @qw estimtorAD provingD in prtiulrD tht the proposed estimtor outperforms the qw estimtorD in the sense tht it hs n symptotilly smller vrineD whers they oth re symptotilly unisedF his n e rgued y the ft thtD in sttistis in generlD the est liner estimtor @or optiml preditorA is sed on the inverse of the utoovrine mtrixD see for instneD fenhenni nd gmnis @IWWPAD wheres the qw estimtor does not tke into ount this orreltion requirementF sn dditionD the qw estimtor is n pproximtion of n integrl ndD s known in sttistisD the est liner pproximtion of n integrl is sed on some projetion propertyF his hpter is orgnized s followsF etion PFP is dedited to rell on the urD whih will e usefull for the onstrution of the new estimtorD in ddition to some tehnil detilsF sn setion PFQD we onstrut our proposed estimtor for the funtion g in wodel @PFIA where ε is enteredD seond order error proess with ontinuous utoovrine funtion RF st is onstruted through the following funtion de(nedD for x ∈ [0, 1]D yD

f x,h (t) = 1 0 R(s, t)ϕ x,h (t) ds where ϕ x,h (t) = 1 h K x -t h for t ∈ [0, 1], @PFPA
where K is uernel nd h = h(n) is ndwidthF e shll see tht this funtion elongs to the ur@RAF his llows us to use the properties of this spe to ontrol the vrine of the proposed estimtorF hese properties were introdued y rzen @IWSWA to solve vrious prolems in sttistil inferene on time seriesF e lso giveD in this setionD the nlytil expressions of this estimtor for the generlised iener proess nd the yrnsteinEhlenek proessD sine the nlytil expression of the inverse of the utoovrine mtrix n e derived for this lss of proessesF sn etion PFRD we derive the symptoti performnes of this estimtorF e give n sympE toti expression of the weights of this liner estimtorD whih is used to derive the symptoti expression of its isF he properties of the ur@RA not only llow us to otin the symptoti expression of the vrineD ut lso to (nd the optiml rte of onvergene of the residul vriE neF efter otining the symptoti expression of the sntegrted wen qured irror @swiAD we derive the symptoti optiml ndwidth with respet to the swi riterionF woreoverD we prove the symptoti normlity of the proposed estimtorF sn etion PFSD we give theoretil omprison etween the new estimtor nd the qsser nd wüller9s estimtorF e prove tht the proposed estimtor hsD symptotillyD smller vrine thn tht of qsser nd wüllerF woreoverD the proposed estimtor hs n symptotilly smller swiD for instneD in the se of iener proess εF sn etion PFTD we ondut simultion study in order to investigte the performne of the proposed estimtor in (nite smple setD then we ompre it with the qsser nd wüller9s estimtor for di'erent vlues of the numer of experimentl units nd di'erent vlues of the RH Chapter 2 smple sizeF ine the lssil rossEvlidtion riterion is shown to e ine0ient in the presene of orreltion @see for instneD eltmn @IWWHAD ghiu @IWVWA nd rrt @IWWID IWWRAD we use the optiml ndwidth tht minimizes the ext swiD otined using the gonjugted qrdient elgorithmF he results of this simultion study on(rm our theoretil sttements given in etion Q nd etion RF pinllyD the supplementry mterils setion is dedited to the proofs of the theoretil resultsF sn the following setionD we introdue the eproduing uernel rilert pesD quik rell out them whih will e useful through out this hpterF 2.2 Reproducing Kernel Hilbert Spaces vet ε = (ε(t)) t∈[0,1] e entered nd seond order proess of utoovrine D suh tht R is invertile when restrited to ny (nite set on [0, 1]. vet L(ε(t), t ∈ [0, 1]) e the set of ll rndom vriles whih mye e written s liner omintions of ε(t) for t ∈ [0, 1]D iFeFD the set of rndom vriles of the form l i=1 α i ε(t i ) for some positive integer l nd some onstnts

α i D t i ∈ [0, 1] for i = 1, • • • , lF vet lso L 2 (ε) e the rilert spe of ll squre integrle rndom vriles in the liner mnifold L(ε(t), t ∈ [0, 1])D together with ll rndom vriles U tht re limits in L 2 of sequene of rndom vriles U n in L(ε(t), t ∈ [0, 1])D iFeD U is suh thtD ∃ (U n ) n≥0 ∈ L(ε(t), t ∈ [0, 1]) : lim n→∞ E((U n -U ) 2 ) = 0.
henote y F(ε) the fmily of funtions g on [0, 1] de(ned yD

F(ε) = {g : [0, 1] → R with g(•) = E(U ε(•)) where U ∈ L 2 (ε)},
e note here tht for every g ∈ F(ε)D the ssoited U is uniqueF st is esy to verify tht F(ε) is rilert spe equipped with the norm || || de(ned for g ∈ F(ε) yD

||g|| 2 = E(U 2 ). sn ftD let g ∈ F(ε)D iFeD g(•) = E(U ε(•)) for some U ∈ L 2 ( )F e hveD • ||g|| = E(U 2 ) ≥ 0. • ||g|| = E(U 2 ) = 0 ⇒ U = 0 FsF ⇒ g = 0. • por g ∈ F(ε)D iFeD f (•) = E(V ε(•)) some V ∈ L 2 ( )F e hveD ||g + f || 2 = E((U + V ) 2 ) = E(U 2 ) + E(V 2 ) + 2E(U V ) ≤ E(U 2 ) + E(V 2 ) + 2 E(U 2 ) E(V 2 ) = E(U 2 ) + E(V 2 ) 2 . husD ||g + f || ≤ E(U 2 ) + E(V 2 ) = ||g|| + ||f ||.
e now prove the ompleteness of

F(ε)F por this let g n (•) = E(U n ε(•)) e guhy sequene in F(ε)D iFeFD lim n,m→∞ ||g n -g m || 2 = 0.
E((U n -U m ) 2 ) = lim n,m→∞ ||g n -g m || 2 = 0.
his yields tht (U n ) n≥1 is guhy sequene in L 2 (ε)D whih is rilert spe s proven y rzen @IWSWA @see pge V thereAF hus it exists U ∈ L 2 (ε) suh thtD

lim n→∞ E((U n -U ) 2 ) = 0. king g(•) = E(U ε(•)), whih is lerly n element of F(ε) givesD lim n→∞ ||g n -g|| 2 = lim n→∞ E((U n -U ) 2 ) = 0.
his onludes the proof of ompletness of F(ε)F he rilert spe F(ε) n esily e identi(ed s the eproduing uernel rilert pe ssoited to reproduing kernel R @with R(s, t) = E(ε(s)ε(t))AD whih is de(ned s followsF Denition 2.2.1 Parzen (1959) A Hilbert space H is said to be a Reproducing Kernel Hilbert Space associated to a reproducing kernel (or function) R (RKHS(R)), if its members are functions on some set T , and if there is a kernel R on T × T having the following two properties:

R(•, t) ∈ H for all t ∈ T, g, R(•, t) = g(t)
for all t ∈ T and g ∈ H, @PFQA where •, • is the inner (or scalar) product in H.

o prove thisD we need to verify the properties given in @PFQAF por t ∈ [0, 1] we hveD

R(s, t) = E(ε(s)ε(t)) for ll s ∈ [0, 1]. ine ε(s) ∈ L 2 (ε) then R(•, t) ∈ F(ε) for ny (xed t ∈ [0, 1]F xow let g ∈ F(ε)D iFeFD g(•) = E(U ε(•)) for some U ∈ L 2 (ε). henD g, R(•, t) = 1 2 ||g|| 2 + ||R(•, t)|| 2 -||g -R(•, t)|| 2 = 1 2 E(U 2 ) + E(ε(t) 2 ) -E((U -ε(t)) 2 ) = 1 2 E(2U ε(t)) = g(t).
hese properties together with the following theorem yield tht F(ε) is the ur@RAF Theorem 2.2.1 (E. H. Moor) Aronszajn (1944) A symmetric non-negative Kernel R generates a unique Hilbert space.

sn the sequelD we tke R to e ontinuous on [0, 1] 2 nd we shll onsider the funtion of interest given y @PFPAF wore generllyD we onsider the funtion f D de(ned for ontinuous funtion

ϕ nd t ∈ [0, 1]D y f (t) = 1 0 R(s, t)ϕ(s) ds. @PFRA Lemma 2.2.1 We have f ∈ F(ε), i.e., there exists X ∈ L 2 (ε) with, f (•) = E(Xε(•)). @PFSA
In addition,

f 2 = E(X 2 ) = 1 0 1 0 R(s, t)ϕ(s)ϕ(t) dt ds. xow let T n = (t 1 , t 2 , • • • , t n ) with 0 ≤ t 1 < t 2 < • • • < t n ≤ 1 nd let V Tn e the suspe of F(ε) spnned y the funtions R(•, t) for t ∈ T n D iFeFD V Tn = {g : [0, 1] → R with g(•) = E(U ε(•)) where U ∈ L(ε(t), t ∈ T n )}. yur tsk is to prove tht if R |Tn = (R(t i , t j ) 1≤i,j≤n ) is nonEsingulr mtrix then V Tn is losed suspe of F(ε)F por this letD (g m ) m≥1 e sequene in V Tn onverging to g ∈ F(ε)F e shll prove tht g ∈ V Tn F xote thtD g m (t) = E(U m ε(t)) with U m = n i=1 a i,m ε(t i ), where (a i,m ) m≥1 ∈ R. ine g m onverges in F(ε) then it is guhy sequeneD iFeFD lim m 1 ,m 2 →∞ ||g m 1 -g m 2 || 2 = 0.
fy the de(nition of the norm on F(ε) we hveD

||g m 1 -g m 2 || 2 = E((U m 1 -U m 2 ) 2 ) = E n i=1 (a i,m 1 -a i,m 2 )ε(t i ) 2 = n i=1 n j=1 (a i,m 1 -a i,m 2 )(a j,m 1 -a j,m 2 )R(t i , t j ) = A m 1 ,m 2 R |Tn A m 1 ,m 2 , where A m 1 ,m 2 = (a 1,m 1 -a 1,m 2 , • • • , a n,m 1 -a n,m 2 ) F husD lim m 1 ,m 2 →∞ A m 1 ,m 2 R |Tn A m 1 ,m 2 = 0. ine R |Tn is symmetri positive mtrixD we otinD lim m 1 ,m 2 →∞ A m 1 ,m 2 = lim m 1 ,m 2 →∞ (a 1,m 1 -a 1,m 2 , • • • , a n,m 1 -a n,m 2 ) = (0, . . . , 0) , whih yields tht (a i,m ) m is guhy sequene on R for ll i = 1, • • • , n. king a i = lim m→∞ a i,m
we otin y the uniqueness of the limitD

g(•) = E(U ε(•)) with U = n i=1 a i ε(t i ),
whih yields tht g ∈ V Tn F rene V Tn is losedF ine V Tn is losed suspe in the rilert spe F(ε)D one n de(ne the orthogonl projetion opertor from F(ε) to V Tn whih we note y P |Tn D iFeFD for every f ∈ F(ε)D

P |Tn f = argmin g ∈V Tn ||f -g||.
xowD for t i ∈ T n D R(•, t i ) ∈ V Tn F reneD for every i = 1, . . . , n. P |Tn f -f, R(•, t i ) = 0 or equivlently P |Tn f, R(•, t i ) = f, R(•, t i ) .
he lst equlityD together with @PFQAD gives thtD

P |Tn f (•) = f (•) on T n . @PFTA
2.3 Construction of the estimator using the RKHS approach e onsider wodel @PFIA where g is the unknown regression funtion on [0, 1] nd {ε j (t), t ∈ [0, 1]} j is sequene of error proessesF e ssume tht g ∈ C 2 ([0, 1]) nd tht (ε j ) j re iFiFdF proesses with the sme distriution s entered seond order proess εF e denote y R its utoovrine funtionD ssumed to e knownD ontinuous nd forms non singulr mtrix when restrited to T × T for ny (nite set T ⊂ [0, 1]F

Projection estimator

sn this setionD we shll give the de(nition of the new proposed estimtor for the regression funtion g in wodel @PFIAF his estimtor @see he(nition PFQFI elowA is onstruted using the funtion f x,h given y @PFPA for

x ∈ [0, 1]D h ∈]0, 1[ nd K is (rst order kernel 1 of support [-1, 1]
elonging to C 1 F his funtion is well known in time series nlysis nd hs een used y severl uthorsF e mentionD mong othersD the work of ks nd lvisker @IWTTA nd of felouni nd fenhenni @PHISA for liner regression models with orrelted errorsF st is minly used due to its elonging to the @ur@AA @see etion PFP for more detilsAF his spe is spnned y the funtions {R(•, t i ) 1≤i≤n } forming losed suspe on whih n orthogonl projetion of the funtion f x,h is fesileF e shll ll the estimtor otined y this pprohD the projetion estimtorF he proposed estimtorD whih is kernel estimtorD is liner in the oservtions Y (t i ) nd is given y the following de(nitionF Denition 2.3.1 The projection estimator of the regression function g in Model @PFIA based on the observations

(t i , Y j (t i )) 1≤i≤n 1≤j≤m is given for any x ∈ [0, 1] by, ĝpro n (x) = n i=1 m x,h (t i )Y (t i ), @PFUA
where

Y (t i ) = 1 m m j=1 Y j (t i ) and the weights (m x,h (t i )) 1≤i≤n are being determined, letting T n = (t i ) 1≤i≤n , by, m x,h |Tn = f x,h |Tn R -1 |Tn , @PFVA with f x,h |Tn := (f x,h (t 1 ), . . . , f x,h (t n )) , R |Tn := (R(t i , t j )) 1≤i,j≤n , R -1
|Tn the inverse of R |Tn and m x,h |Tn := (m x,h (t 1 ), . . . , m x,h (t n )) , where v denotes the transpose of a vector v. 1 The kernel K satises:

1 -1 K(t)dt = 1, 1 -1 tK(t)dt = 0 and 1 -1 t 2 K(t)dt < +∞.

RR Chapter 2

Remark 2.3.1 In order to motivate the proposed estimator, consider the regression model using m continuous experimental units, i.e.,

Y j (t) = g(t) + ε j (t) for t ∈ [0, 1] and j = 1, • • • , m. @PFWA
A continuous kernel estimator of g in Model @PFWA is given for any

x ∈ [0, 1] by, ĝ[0,1] (x) = 1 0 ϕ x,h (t)Y (t) dt with Y (t) = 1 m m j=1 Y j (t), @PFIHA where ϕ x,h (t) = 1 h K x-t h
for a kernel K and a bandwidth h. We refer the reader to the works of Blanke and Bosq ( 2008) and Didi and Louani (2013) for more details on the Kernel estimation of the regression function based on continuous observations. Since in practice we only have access to discrete observations, then a linear approximation of the continuous kernel estimator should be of the form:

ĝn (x) = n i=1 W x,h (t i )Y (t i ). Now let, f n,x (t) = n i=1 W x,h (t i )R(t i , t) for t ∈ [0, 1].
Then the Mean Squared Error (MSE) of approximation can be written as:

E ĝ[0,1] (x) -ĝn (x) 2 = ||f x,h -f n,x || 2 ,
where f x,h is given by @PFPA and || • || is the norm of the RKHS(R)(see Section 2.2 for more details). Then the best linear predictor ĝpro n (x) of ĝ[0,1] (x) satises:

inf W x,h |Tn E ĝ[0,1] (x) -ĝn (x) 2 = ||f x,h -P |Tn f x,h || 2 ,
where P |Tn f x,h is the orthogonal projection of f x,h on the subspace of RKHS spanned by the function

{R(•, t i ), i = 1, • • • , n}.
The optimal coecients (W * x,h (t i )) 1≤i≤n can then be derived by using the fact that

P |Tn f x,h (t i ) = f x,h (t i ) for i = 1, • • • , n (see Equation @PFTA) and this yields W * x,h |Tn = f x,h |Tn R -1 |Tn .
por some lssil error proessesD suh s the iener nd the yrnsteinEhlenek proessesD the estimtor @PFUA hs simpli(ed expression s shown in the following propositionF Proposition 2.3.1 Consider the regression model @PFIA where ε is of autocovariance function

R(s, t) = min(s,t) 0 u β du for a positive constant β. Let t 0 = 0, t n+1 = 1. Set Y (t 0 ) = 0 and Y (t n+1 ) = Y (t n ).
For any x ∈ [0, 1], the projection estimator @PFUA can be written as follows:

ĝpro n (x) = 1 β + 1 n+1 i=1 Y (t i ) t i t i-1 ϕ x,h (s)ds + n-1 i=0 Y (t i+1 ) -Y (t i ) t β+1 i+1 -t β+1 i t i+1 t i (s β+1 -t β+1 i+1 )ϕ x,h (s)ds . @PFIIA
Remark 2.3.2 Taking β = 0 in the previous proposition gives the expression of the projection estimator @PFUA in the case where ε is the classical standard Wiener error process.

Proposition 2.3.2 If the error process ε in Model @PFIA is the Ornstein-Uhlenbeck process with R(s, t) = e -|t-s| then for any x ∈ [0, 1],

ĝpro n (x) = n-1 i=2 Y (t i ) t i+1 t i-1 e |s-t i | ϕ x,h (s) ds + Y (t 1 ) t 2 0 e s-t 1 ϕ x,h (s) ds + Y (t n ) 1 t n-1 e tn-s ϕ x,h (s) ds - n-1 i=1 e t i+1 Y (t i+1 ) -e t i Y (t i ) 1 -e -2(t i+1 -t i ) t i+1 t i e -s ϕ x,h (s) ds + n-1 i=1 e -t i+1 Y (t i+1 ) -e -t i Y (t i ) 1 -e -2(t i+1 -t i ) t i+1 t i e s ϕ x,h (s) ds,
where ϕ x,h is dened in the previous proposition.

Remark 2.3.3 As the previous propositions show, the expression of m x,h |Tn is known analytically for error processes of practical interest. For more complicated error processes, numerical methods can be used. For more general error processes, we will give an asymptotic simplied expression of the weights of the projection estimator (see Lemma 2.4.2 below).

Assumptions and comments

sn order to derive our symptoti resultsD the following ssumptions on the utoovrine funE tion R nd the uernel K re requiredF @eA R is ontinuous on the entire unit squre nd hs left nd right derivtives up to order two t the digonl @iFeF when s = tAD iFeFD

R (0,1) (t, t -) = lim s↑t ∂R(t, s) ∂s nd R (0,1) (t, t + ) = lim s↓t ∂R(t, s) ∂s ,
exist nd re ontinuousF sn similr wy we de(ne R (0,2) (t, t -) nd R (0,2) (t, t + )F y' the digonl @iFeF when s = t in the unit squreAD R hs ontinuous derivtives up to order twoF o otin our symptoti resultsD we shll give next stronger ssumption on the jump funtion αF

por t ∈]0, 1[D let α(t) = R (0,1) (t, t -) -R (0,1) (t, t + )F essumption @
@fA e ssume tht α is vipshitz on ]0, 1[D inf 0<t<1 α(t) = α 0 > 0 nd sup 0<t<1 α(t) = α 1 < ∞F
essumptions @eA nd @fA re lssil regulrity onditions nd were used in severl worksD see for instneD ks nd lvisker @IWTTAD u nd gmnis @IWWQA nd most reently felouni nd fenhenni @PHISAF @gA por eh t ∈ [0, 1]D R (0,2) (., t + ) is in the eproduing uernel rilert spe ssoited to RD denoted y ur@RAD equipped with the norm || • ||F sn dditionD sup 0≤t≤1 ||R (0,2) (., t + )|| < ∞ @see etion PFP for more detilsAF essumption @gAD whih is more restritive thn @fA s indited y ks nd lvisker @IWTTAD is neessry to evlute the weights of the projetion estimtor @see vemm PFRFP elowAF @hA K is n even funtion nd K is vipshitz funtion on [-1, 1]F ixmples of utoovrine funtions whih stisfy essumptions (A)D (B) nd (C) re given elowF

Example 2.3.1

1. The autocovariance function R(s, t) = σ 2 min(s, t) of the Wiener process, has a constant jump function α(t) = σ 2 and R (i,j) (s, t) = 0 for all integers i, j such that i + j = 2 and

s = t.
2. The autocovariance function R(s, t) = σ 2 e -λ|s-t| of the stationary Ornstein-Uhnelbeck process with σ > 0 and λ > 0. For this process the jump function is

α(t) = 2σ 2 λ and R (0,2) (s, t) = σ 2 λ 2 e -λ|s-t| .
3. Another general class of autocovariance functions was given by Sacks and Ylvisaker (1966) and has the form,

R(s, t) = 1/|t-s| 0 (1 -µ|t -s|)p(µ) dµ,
where p is a probability density and p its derivative are such that,

lim µ→∞ µ 3 p(µ) < ∞, and ∞ a (µp (µ) + 3p(µ)) 2 )µ 6 dµ < ∞,
for some a. We have α(t) = 2 ∞ 0 up(u) du.

Local asymptotic results

vet T n = (t i,n ) 1≤i≤n for n ≥ 1D e (xed sequene of designs with T n ∈ D n D whereD

D n = {(s 1 , s 2 , . . . , s n ) : 0 ≤ s 1 < s 2 < • • • < s n ≤ 1}. et t 0,n = 0, t n+1,n = 1D d j,n = t j+1,n -t j,n nd let for x ∈ [0, 1]D h = h(n)D I x,h = {i = 1, • • • , n : [t i-1,n , t i+1,n ]∩]x -h, x + h[ = ∅}. henote y N Tn = Card(I x,h ). ell tht [x -h, x + h]
is the support of the funtion ϕ x,h F o otin the symptoti resultsD we require tht the sequene (T n ) n≥1 stis(es the next ssumptionF

@iA lim n→∞ sup 0≤j≤n d j,n = 0D lim n→∞ 1 h sup 0≤j≤n d j,n = 0D lim n→∞ N Tn 1 h 2 sup 0≤j≤n d 2 j,n = 0 nd lim sup n→∞ N 2 Tn 1 h 2 sup 0≤j≤n d 2 j,n < ∞ F 2.4. LOCAL ASYMPTOTIC RESULTS

RU

e simple sequene of designs tht veri(es essumption (E) ws presented y ks nd lvisker @IWUHA s followsF Denition 2.4.1 Let F be a distribution function of some density function f such that sup

0<t<1 f (t) < ∞ and inf 0<t<1 f (t) > 0.
The so-called regular sequence of designs generated by f is dened by,

T n = t i,n = F -1 i n , i = 1, . . . , n .
sn the sequelD the density f is ssumed to e t lest in C 2 ([0, 1])F his sequene of designs veri(es the following vemm @see for instne fenelmdni et al. @PHIWA for its proofAF Lemma 2.4.1 Let (T n ) n≥1 be a regular sequence of designs generated by some density function.

For x ∈]0, 1[ and h > 0, suppose that T n ∩ [x -h, x + h] = ∅ and that nh ≥ 1. Then, sup 0≤j≤n d j,n = O 1 n and N Tn = O(nh), @PFIPA
where N Tn and d j,n are dened as above. In addition, if lim n→∞ nh = ∞ then the regular sequence of designs veries Assumption (E).

Evaluation of the bias

sn order to derive the symptoti expression of the is term of the projetion estimtorD we shll (rst give the symptoti pproximtion of the weights m x,h |Tn @de(ned y @PFVAA in the following lemmF Lemma 2.4.2 Suppose that Assumptions (A), (B) and (C) are satised. Then for any x ∈]0, 1[,

m x,h (t i,n ) =            1 2 ϕ x,h (t i,n )(t i+1,n -t i-1,n ) + O α n,h + β n,h if i / ∈ {1, n} and [t i-1,n , t i+1,n ] ∩ [x -h, x + h] = ∅, O N Tn α n,h + nβ n,h if i ∈ {1, n}, O β n,h otherwise,
where,

α n,h = sup 0≤i≤n sup t i,n ≤s,t≤t i+1,n d i,n |α(s)ϕ x,h (s) -α(t)ϕ x,h (t)| = O 1 h 2 sup 0≤j≤n d 2 j , β n,h = sup 0≤t≤1 1 α(t) ||R (0,2) (., t)|| √ C √ h sup 0≤j≤n d 2 j = O 1 √ h sup 0≤j≤n d 2 j ,
and C is a positive constant dened in Proposition 2.4.3 below.

Remark 2.4.1 This Lemma shows that the weights of the projection estimator are asymptoticly equivalent to those of some well known linear estimators of the regression function g. For instance,

• Priestly and Chao (1972) used the following weights:

W x,h (t i ) = (t i+1,n -t i,n )ϕ x,h (t i ) for i = 1, • • • , n.
• Gasser and Müller (1979) used the following weights:

W x,h (t i ) = s i,n s i-1,n ϕ x,h (s) ds for i = 1, • • • , n,
where, s 0 = 0, s n = 1 and s

i,n = (t i+1,n + t i,n )/2 for i = 1, • • • , n -1.
• Cheng and Lin (1981) replaced s i,n by t i,n , in the weights of the Gasser and Müller estimator.

sing the symptoti pproximtion of the weights given in vemm PFRFPD we n otin the symptoti expression of the is of the projetion estimtor s shows the following propositionF Proposition 2.4.1 Suppose that Assumptions

(A) -(D) are satised. If T n ∩]x -h, x + h[ = ∅ and nh ≥ 1, then for any x ∈]0, 1[, E(ĝ pro n (x)) -g(x) = 1 2 h 2 g (x)B + o(h 2 ) + O N Tn h 3 sup 0≤j≤1 d 3 j,n + N Tn α n,h + nβ n,h ,
where α n,h and β n,h are given in Lemma 2.4.2 and B = 1

-1 t 2 K(t) dt.
Remark 2.4.2 Under the assumption of Lemma 2.4.1 we have,

E (ĝ pro n (x)) -g(x) = 1 2 h 2 g (x)B + o(h 2 ) + O 1 nh .
sn the se of iener error proessD diret omputtion of the is term of the projetion estimtor @PFIIAD with β = 0D shows tht the order term O 1 nh n e improvedF he result is given y the following propositionF Proposition 2.4.2 Consider Model @PFIA with a Wiener error process of autocovariance function R(s, t) = min(s, t). Let (T n ) n≥1 be a regular sequence of designs generated by a density function f (cf. Denition 2.4.1) and let K be a kernel satisfying Assumption (D).

If T n ∩]x -h, x + h[ = ∅ and nh ≥ 1 then, E (ĝ pro n (x)) -g(x) = 1 2 h 2 g (x)B + o(h 2 ) + O 1 n 2 h ,
where B is given in Proposition 2.4.1 above.

Evaluation of the variance

st is shown in vemm PFPFI of etion PFP tht f x,h de(ned y @PFPA elongs to the ur@RA equipped with its norm || || ndD

||f x,h || 2 = 1 0 1 0 ϕ x,h (s)R(s, t)ϕ x,h (t)ds dt ∆ = σ 2 x,h . @PFIQA sn ddition if P |Tn f x,h
is the projetion of f x,h on the suspe of F spnned y {R(., t), t ∈ T n } then it is shown y @pPA in the supplementry fts of the eppendix thtD

||P |Tn f x,h || 2 = mVar ĝpro n (x
K ∞ = sup t∈[-1,1] |K(t)|, R 1 = sup t,s∈[0,1] |R (1,1) (s-, t+)| and R 2 = sup t,s∈[0,1] |R (0,2) (s, t+)|.
Then we have, for any

x ∈]0, 1[, 0 ≤ σ 2 x,h m -Var ĝpro n (x) ≤ C mh sup 0≤j≤n d 2 j,n , where C = K 2 ∞ ( 4 3 α 1 + R 1 + 4 3 R 2 ) if (x -h) and (x + h) ∈ T n , K 2 ∞ ( 8 3 α 1 + 5 3 R 1 + 8 3 R 2 )
otherwise.

(x) ≥ 1 12 α(x) 1 -1 K 2/3 (t)dt 3 , @PFISA
where σ 2 x,h is given by @PFIQA.

sing ropositions PFRFQ nd PFRFR we n otin the optiml onvergene rte 1/(mn 2 h) of the residul vrineF he result is given y the following propositionF Proposition 2.4.5 Suppose that all the assumptions of Lemma 2.4.1, Propositions 2.4.3 and 2.4.4 are satised. Then there exist some positive constants C and C such that for any x ∈]0, 1[ and for any positive integer m,

lim n→∞ mn 2 h σ 2 x,h m -Var(ĝ pro n (x)) ≤ C, @PFITA
nh = ∞ then for any x ∈]0, 1[, Var(ĝ pro n (x)) = σ 2 x,h m - 1 12mn 2 x+h x-h α(t) f 2 (t) ϕ 2 x,h (t)dt + O 1 mn 3 h 2 , @PFIVA
where σ 2 x,h is given by @PFIQA.

he following lemm @proved in fenhenni nd hdi @PHHUA gives the expression of the min term of the symptoti vrine σ 2 x,h /m in terms of hF Lemma 2.4.3 Suppose that Assumptions (A), (B) and (D) are satised. If

lim n→∞ h = 0 then, for any x ∈]0, 1[, σ 2
x,h (as given by @PFIQA) has the following asymptotic expression

σ 2 x,h = R(x, x) - 1 2 α(x)C K h + o(h), @PFIWA
where

C K = 1 -1 1 -1 |u -v|K(u)K(v)dudv.

IMSE and optimal bandwidth

roposition PFRFT nd emrk PFRFP llow to derive the symptoti expression of the wen qured irror @MSEA nd the sntegrted wen qured irror @IMSEA of the projetion estiE mtor @PFUA givenD without proofD in the next theoremF Theorem 2.4.1 If all the assumptions of Propositions 2.4.1 and 2.4.6 are satised and if (T n ) n≥1 is a regular sequence of designs generated by some density function (see Denition 2.4.1) then for any

x ∈]0, 1[, MSE(ĝ pro n (x)) = 1 m R(x, x) - 1 2 α(x)C K h + 1 4 h 4 (g (x)) 2 B 2 + o h 4 + h m + O 1 mn 2 h + h n + 1 n 2 h 2 , IMSE(ĝ pro n ) = 1 m 1 0 R(x, x)w(x) dx - C K h 2m 1 0 α(x)w(x) dx + B 2 4 h 4 1 0 [g (x)] 2 w(x) dx + o h 4 + h m + O 1 mn 2 h + h n + 1 n 2 h 2 ,
where w is a positive density function, B is given in Proposition 2.4.1 and C K is given in Lemma 2.4.3.

Remark 2.4.3 We note here that the term 1

12mn 2 x+h x-h α(t) f 2 (t) ϕ 2
x,h (t)dt appearing in the asymptotic variance, does not appear in the asymptotic MSE and IMSE, because it is negligible comparing to the squared bias, precisely due to the term O 1 nh . However in the case of a Wiener error process, we have proven (see Proposition 2.4.2) that the previous term can be replaced by O 1 n 2 h when using exact weights of the projection estimator (and not their asymptotic expression). Therefor, when ε is a Wiener process, the asymptotic expressions of the MSE and IMSE of the projection estimator @PFIIA (with β = 0) are given by the following theorem.

Theorem 2.4.2 Consider Model @PFIA with a Wiener error process and suppose that the kernel K veries Assumption (D). Moreover, assume that (T n ) n≥1 is a regular sequence of designs generated by a function f (see Denition 2.4.1). If lim n→∞ h = 0 and lim n→∞ nh = ∞ then for any

x ∈]0, 1[, MSE(ĝ pro n (x)) = 1 m R(x, x) - 1 2 α(x)C K h - 1 mn 2 h α(x) f 2 (x) 1 -1 K 2 (t) dt + 1 4 h 4 [g (x)] 2 B 2 + o h m + h 4 + O h n 2 + 1 mn 3 h 2 + 1 mn 2 + 1 n 4 h 2 ,
and,

IMSE(ĝ pro n ) = 1 m 1 0 R(x, x)w(x) dx - C K h 2m 1 0 α(x)w(x) dx - A 12mn 2 h 1 0 α(x) f 2 (x) w(x) dx + B 2 4 h 4 1 0 [g (x)] 2 w(x) dx + o h m + h 4 + O h n 2 + 1 mn 3 h 2 + 1 mn 2 + 1 n 4 h 2 , where A = 1 -1 K 2 (t) dt, w, B and C K are given in Theorem 2.4.1.
he symptoti optiml ndwidth is otined y minimizing the symptoti swi nd is given in the following orollryF Corollary 2.4.1 (Optimal bandwidth) Suppose that the assumptions of Theorem 2.4.1 are satised. Moreover assume that n m = O(1) as n, m → ∞. Denote by IMSE(h) the IMSE of the projection estimator when the bandwidth h is used. Then the bandwidth,

h * = C K 1 0 α(x)w(x) dx 2B 1 0 [g (x)] 2 w(x) dx 1/3 m -1/3 , @PFPHA
is optimal in the sense that,

lim n,m→∞ IMSE(h * ) IMSE(h n,m ) ≤ 1,
for any sequence of bandwidths h n,m verifying: √ mh 2 = 0. Then for any

x ∈]0, 1[, √ m ĝpro n (x) -g(x) D -→ Z with Z ∼ N (0, R(x, x)) as n, m → ∞,
where D denotes the convergence in distribution and N is the normal distribution. 2.5 Comparison with the Gasser and Müller's estimator sn this setionD we shll perform theoretil omprison etween the projetion estimtor given in @PFUA nd the lssil estimtor proposed y qsser nd wüller @IWUWA tht we rell in the de(nition elowF Denition 2.5.1 The Gasser and Müller's estimator of the regression function g based on the observations

(t i , Y j (t i )) 1≤i≤n 1≤j≤m is given for any x ∈ [0, 1] by, ĝGM n (x) = n i=1 Y (t i ) s i s i-1 ϕ x,h (s) ds , @PFPIA
where Y , ϕ x,h and h are given in Denition 2.3.1. The midpoints (s i ) 1≤i≤n are such that:

s 0 = 0, s n = 1 and for i = 1, . . . , n -1, s i = (t i + t i+1 )/2.
sn order to ompre this estimtor to the projetion estimtor with respet to the swiD we rell in the next theorem the symptoti expression of the swi of the qsser nd wüller9s estimtor @given in ghpter IAF

nh = ∞ then for any x ∈]0, 1[, MSE(ĝ GM n (x)) = 1 m R(x, x) - 1 2 α(x)C K h + 1 4 h 4 (g (x)) 2 B 2 + o h 4 + h m + O h n 2 + 1 n 4 h 2 + 1 mn 3 h 2 + 1 mn 2 ,
and,

IMSE(ĝ GM n ) = 1 m 1 0 R(x, x)w(x) dx - C K h 2m 1 0 α(x)w(x) dx + B 2 4 h 4 1 0 [g (x)] 2 w(x) dx + o h 4 + h m + O h n 2 + 1 n 4 h 2 + 1 mn 3 h 2 + 1 mn 2 ,
where B and C K are given in Propositions 2.4.1 and 2.4.6 and w is a continuous positive density.

he following theorem gives n symptoti omprison in term of the vrine of the projetion estimtor @PFUA nd the qsser nd wüller9s estimtor @PFPIAF Theorem 2.5.2 Suppose that Assumptions (A), (B) and (D) are satised. Moreover assume that (T n ) n≥1 is a regular sequence of designs generated by a density function f (see Denition 2.4.1). If

lim n→∞ h = 0 and lim n→∞ nh = ∞ then for any x ∈]0, 1[, lim n,m→∞ mn 2 h Var ĝGM n (x) -Var ĝpro n (x) = 1 12 α(x) f 2 (x) > 0.
por omprison of the is of these estimtorsD we mention tht the qsser nd wüller9s estimtor onverges to zero slightly fster thn the is of the projetion estimtorD iFeFD the term O( 1 nh ) in the is of the projetion estimtor @see emrk PFRFPA is repled y O( 1 n 2 h ) in the is of the qsser nd wüller9s estimtor @see roposition IFQFIAF roweverD for the iener error proess oth estimtors hve the sme is onvergene rtesD thus we n ompre the symptoti swi of oth estimtors in the following theoremF Theorem 2.5.3 Consider Model @PFIA where ε is a Wiener error process. Suppose that the assumptions of Theorem 2.4.1 are satised. Moreover, assume that lim n→∞ nh 2 = 0 and that m n = O(1) then,

lim n,m→∞ mn 2 h (IMSE (ĝ GM n ) -IMSE (ĝ pro n )) = σ 2 12 1 0 w(x) f 2 (x) dx > 0.
Remark 2.5.1 Theorems 2.5.2 and 2.5.3 show that, the projection estimator has an asymptotically smaller variance than the Gasser and Müller's estimator for any error process, it also has an asymptotically smaller IMSE when ε is a Wiener error process. However the Gasser and Müller's estimator doesn't require the knowledge of the autocovariance function whereas the projection estimator does.

Simulation study

sn this setionD we investigte the performne of the proposed estimtor @PFUA using (nite vlues of experimentl units m nd smpling points nF he following growth urves re onsideredX @wIA g(x) = 10x 3 -15x 4 + 6x 5 for 0 < x < 1.

@wPA g(x) = x + 0.5 e -80(x-0.5) 2 for 0 < x < 1.

his growth urves were used y rrt nd herly @IWVTA nd fenhenni nd hdi @PHHUA due to its similrity in shpe to tht of the logisti funtionD whih is frequently found in growth urve nlysis s noted y rrt nd herly @IWVTAF he smpling points re tken to eX

t i = (i -0.5)/n for i = 1, • • • , n. @PFPPA
he error proess ε is tken to e the iener error proess with utoovrine funtion R(s, t) = σ 2 min(s, t)F he uernel used here is the qurti kernel given y K(u) = 15 16 (1 -u 2 ) 2 I [-1,1] (u) nd the ndwidth is the optiml one with respet to the ext IMSED otined using the gonjugted qrdient elgorithm @gqeAF e onsider the men of ll estimtors otined from IHH simultionsF e tke σ 2 = 0.5D simultions for other vlues of σ 2 gve similr resultsF he results re given in pigures PFI nd PFP for wodels @wIA nd @wPA respetivelyD for (xed numer of oservtions n = 100 nd three di'erent vlues of experimentl units m = 5, 20, 50F SR Chapter 2 pigure PFIX he regression funtion of model @wIA is in plin line nd the projetion estimtor is in dshed lineF e n see for wodel @wIAD from pigure PFID tht the projetion estimtor gets loser to the regression funtion when m gets iggerD whih proves its good performne nd onsisteny when m inresesF hese results re on(rmed for the growth urve wodel @wPA given in pigure PFPF pigure PFPX he regression funtion of model @wPA is in plin line nd the projetion estimtor is in dshed lineF sn this simultion studyD we onsider the omprison of the proposed estimtor @PFUA to the qsser nd wüller @PFPIA @referred y qw estimtorA with respet to the ext swi in (nite smple setF por thisD we onsider the ui growth urve of model @wIAF e onsider lso the uniform design given y @PFPPA nd the qurti kernel

K(u) = 15 16 (1 -u 2 ) 2 I [-1,1] (u)F

IMSE(h)F

he ndwidth h is hosen through the lgorithm gqeF he results re given in les PFI nd PFP for n = 10 nd for di'erent vlues of mF hese tles present the integrted is squred denoted y Ibias 2 D integrted vrine denoted y Ivar nd the IMSE together with the optiml ndwidth ssoited to eh estimtorF SS pirstD we n see from these two tles thtD the optiml ndwidth dereses when m inresesD s shown in gorollry PFRFIF sn dditionD the optiml ndwidth of the projetion estimtor is slightly smller thn tht of the qw estimtorF st is lso seen tht oth the Ivar nd the Ibias 2 D of the two estimtors derese when m inresesF sn dditionD the projetion estimtor hs smller Ibias 2 nd Ivar thn tht of the qw estimtorD whih leds to smller IMSEF enother wy to look t these results is s followsX for (xed numer of experimentl units m = 10 nd when the error proess is iener proess @similr results for the yrnsteinEhlenek error proessAD the projetion estimtor would only need n = 10 oservtions on eh experiE mentl unit to otin the performne IMSE = 4.53 × 10 -2 @see le PFIAD wheres the qw estimtor would need to hve n = 18 oservtions to otin the sme performneD nd thus requires 80% more smples in order to hieve the sme performneF he results of this simultion study show thtD even for smll numer of oservtionsD the projetion estimtor outperforms the qw estimtor with respet to swiF st should e noted here thtD in order to solve the prolem t the edges [0, h] ∩ [1 -h, 1]D it ws neessry to djust the kernel s suggested y rrt nd herly @IWVTAF le PFIX he integrted squred isD integrted vrineD swi nd the optiml ndwidth for n = 10 nd di'erent vlues of m under the iener error proessD for the qw nd the projetion estimtorsF

n =
(x i,n ) i=1,••• ,n of [0, 1]D X n = n-1 i=1 (x i+1,n -x i,n )ϕ(x i,n )ε(x i,n ) ∈ L 2 (ε), suh tht for ny t ∈ [0, 1]D f (t) = lim n→∞ n-1 i=1 (x i+1,n -x i,n )ϕ(x i,n )R(x i,n , t) = lim n→∞ E(X n ε(t)). e shll prove tht (X n ) n onverges to ertin element of L 2 D iFeFD ∃ X ∈ L 2 : lim n→∞ E (X n -X) 2 = 0, @PFPQA
nd y the de(nition of L 2 (ε) the limit in @PFPQA proves tht X is n element of L 2 (ε)F xow the proof @PFPQA is immediteD in ft it is esy to hek tht (X n ) id guhy sequene in L 2 F fy the ompleteness of L 2 D we dedue @PFPQAF sn ddition we hveD lim n→∞ E(X n ε(t)) = E(Xε(t)), this is due to the following inequlityD

E(X n ε(t)) -E(Xε(t)) ≤ E (X n -X)ε(t) ≤ E((X n -X) 2 ) E(ε(t) 2 ), 2.7. PROOFS SU nd the ft tht lim n→∞ E((X n -X) 2 ) = 0 nd E(ε(t) 2 ) < ∞F he proof of @PFSA is onludedF pinllyD E(X 2 ) = lim n→∞ E(X 2 n ) = lim n→∞ n i=1 n j=1 (x i+1,n -x i,n )(x j+1,n -x j,n )ϕ(x i,n )ϕ(x j,n )R(x i,n , x j,n ) = 1 0 1 0 ϕ(t)ϕ(t)R(s, t) ds dt.
his onludes the proof of vemm PFPFIF Proof of Proposition 2.3.1.

st is known tht @seeD for instne u nd gmnis @IWWQA pge VVA if R(s, t) = min(s,t) 0 u β du then for ny funtions u nd v nd for ny smpling design T n we hveD

u |Tn R -1 |Tn v |Tn = u(t 1 )v(t 1 ) t β+1 1 + n-1 k=1 (u(t k+1 ) -u(t k ))(v(t k+1 ) -v(t k )) t β+1 k+1 -t β+1 k . epling u = f x,h nd v = Y we hveD ĝpro n (x) = f x,h (t 1 )Y (t 1 ) t β+1 1 + n-1 i=1 (f x,h (t i+1 ) -f x,h (t i ))(Y (t i+1 ) -Y (t i )) t β+1 i+1 -t β+1 i . ell tht R(s, t) = 1 β+1 min(s, t) β+1 ndD f x,h (t i ) = 1 0 R(s, t i )ϕ x,h (s) ds = 1 β + 1 t i 0 s β+1 ϕ x,h (s) ds + t β+1 i 1 t i ϕ x,h (s) ds . husD f x,h (t i+1 ) -f x,h (t i ) = 1 β + 1 t i+1 0 s β+1 ϕ x,h (s) ds + t β+1 i+1 1 t i+1 ϕ x,h (s) ds - t i 0 s β+1 ϕ x,h (s) ds -t β+1 i 1 t i ϕ x,h (s) ds + t β+1 i+1 1 t i ϕ x,h (s) ds -t β+1 i+1 1 t i ϕ x,h (s) ds = 1 β + 1 t i+1 t i (s β+1 -t β+1 i+1 )ϕ x,h (s) ds + (t β+1 i+1 -t β+1 i ) 1 t i ϕ x,h (s) ds . husD ĝpro n (x) = f x,h (t 1 )Y (t 1 ) t β+1 1 + 1 β + 1 n-1 i=1 (Y (t i+1 ) -Y (t i )) 1 t i ϕ x,h (s) ds + n-1 i=1 Y (t i+1 ) -Y (t i ) t β+1 i+1 -t β+1 i t i+1 t i (s β+1 -t β+1 i+1 )ϕ x,h (s) ds = f x,h (t 1 )Y (t 1 ) t β+1 1 + 1 β + 1 n-1 i=2 Y (t i ) t i t i-1 ϕ x,h (s) ds -Y (t 1 ) 1 t 1 ϕ x,h (s) ds + Y (t n ) 1 t n-1 ϕ x,h (s) ds + n-1 i=1 Y (t i+1 ) -Y (t i ) t β+1 i+1 -t β+1 i t i+1 t i (s β+1 -t β+1 i+1 )ϕ x,h (s) ds .

SV

Chapter 2

vetting t 0 = Y (t 0 ) = 0 we hveD

f x,h (t 1 )Y (t 1 ) t β+1 1 = 1 β + 1 Y (t 1 ) t β+1 1 t 1 0 s β+1 ϕ x,h (s) ds + Y (t 1 ) 1 t 1 ϕ x,h (s) ds = 1 β + 1 Y (t 1 ) -Y (t 0 ) t β+1 1 -t β+1 0 t 1 0 (s β+1 -t β+1 1 )ϕ x,h (s) ds + Y (t 1 ) t 1 0 ϕ x,h (s) ds + Y (t 1 ) 1 t 1 ϕ x,h (s) ds . pinllyD ĝpro n (x) = 1 β + 1 n-1 i=2 Y (t i ) t i t i-1 ϕ x,h (s) ds -Y (t 1 ) 1 t 1 ϕ x,h (s) ds + Y (t n ) 1 t n-1 ϕ x,h (s) ds + Y (t n ) tn t n-1 ϕ x,h (s) ds -Y (t n ) tn t n-1 ϕ x,h (s) ds + n-1 i=1 Y (t i+1 ) -Y (t i ) t β+1 i+1 -t β+1 i t i+1 t i (s β+1 -t β+1 i+1 )ϕ x,h (s) ds + Y (t 1 ) -Y (t 0 ) t β+1 1 -t β+1 0 t 1 0 (s β+1 -t β+1 1 )ϕ x,h (s) ds + Y (t 1 ) t 1 0 ϕ x,h (s) ds + Y (t 1 ) 1 t 1 ϕ x,h (s) ds = 1 β + 1 n+1 i=1 Y (t i ) t i t i-1 ϕ x,h (s) ds + n-1 i=0 Y (t i+1 ) -Y (t i ) t β+1 i+1 -t β+1 i t i+1 t i (s β+1 -t β+1 i+1 )ϕ x,h (s) ds ,
where

t n+1 = 1 nd Y (t n+1 ) = Y (t n )F his onludes the proof of roposition PFQFIF Proof of Proposition 2.3.2.
st is known @see enderson @IWTHA pge PIHA tht for every funtions u nd v nd for every design T n we hveD

u |Tn R -1 |Tn v |Tn = u(t 1 )v(t 1 ) 1 -e -2(t 2 -t 1 ) + u(t n )v(t n ) 1 -e -2(tn-t n-1 ) + n-1 i=2 u(t i )v(t i )(1 -e -2(t i+1 -t i-1 ) ) (1 -e -2(t i+1 -t i ) )(1 -e -2(t i -t i-1 ) ) - n-1 i=1 u(t i )v(t i+1 ) + u(t i+1 )v(t i ) 1 -e -2(t i+1 -t i ) e -(t i+1 -t i ) . king u = f x,h nd v = Y we getD ĝpro n (x) = f x,h (t 1 )Y (t 1 ) 1 -e -2(t 2 -t 1 ) + f x,h (t n )Y (t n ) 1 -e -2(tn-t n-1 ) + n-1 i=2 f x,h (t i )Y (t i )(1 -e -2(t i+1 -t i-1 ) ) (1 -e -2(t i+1 -t i ) )(1 -e -2(t i -t i-1 ) ) - n-1 i=1 f x,h (t i )Y (t i+1 ) + f x,h (t i+1 )Y (t i ) 1 -e -2(t i+1 -t i ) e -(t i+1 -t i ) ∆ = f x,h (t 1 )Y (t 1 ) 1 -e -2(t 2 -t 1 ) + f x,h (t n )Y (t n ) 1 -e -2(tn-t n-1 ) + A. @PFPRA 2.7. PROOFS SW xote thtD 1 -e -2(t i+1 -t i-1 ) = (1 -e -2(t i+1 -t i ) ) + (1 -e -2(t i -t i-1 ) ) -(1 -e -2(t i -t i-1 ) )(1 -e -2(t i+1 -t i ) ). husD A = n-1 i=2 f x,h (t i )Y (t i ) 1 -e -2(t i -t i-1 ) + n-1 i=2 f x,h (t i )Y (t i ) 1 -e -2(t i+1 -t i ) - n-1 i=2 f x,h (t i )Y (t i ) - n i=2 f x,h (t i-1 )Y (t i ) 1 -e -2(t i -t i-1 ) e -(t i -t i-1 ) - n-1 i=1 f x,h (t i+1 )Y (t i ) 1 -e -2(t i+1 -t i ) e -(t i+1 -t i ) = n-1 i=2 Y (t i ) 1 -e -2(t i -t i-1 ) f x,h (t i ) -f x,h (t i-1 )e -(t i -t i-1 ) - f x,h (t n-1 )Y (t n ) 1 -e -2(tn-t n-1 ) e -(tn-t n-1 ) + n-1 i=2 Y (t i ) 1 -e -2(t i+1 -t i ) f x,h (t i ) -f x,h (t i+1 )e -(t i+1 -t i ) - f x,h (t 2 )Y (t 1 ) 1 -e -2(t 2 -t 1 ) e -(t 2 -t 1 ) - n-1 i=2 f x,h (t i )Y (t i ) @PFPSA imple lultions yieldD f x,h (t i ) -f x,h (t i-1 )e -(t i -t i-1 ) = e -t i t i t i-1 e s ϕ x,h (s) ds -e t i t i t i-1 e -s ϕ x,h (s) ds + e t i (1 -e -2(t i -t i-1 ) ) 1 t i-1
e -s ϕ x,h (s) ds. @PFPTA sn the sme wy we hveD

f x,h (t i ) -f x,h (t i+1 )e -(t i+1 -t i ) = e t i t i+1 t i e -s ϕ x,h (s) ds -e -t i t i+1 t i e s ϕ x,h (s) ds + e -t i (1 -e -2(t i+1 -t i ) ) t i+1 0 e s ϕ x,h (s) ds.
@PFPUA st is esy to verify thtD

n-1 i=2 f x,h (t i )Y (t i ) = n-1 i=2 Y (t i )e -t i t i 0 e s ϕ x,h (s) ds + n-1 i=2 Y (t i )e t i 1 t i e -s ϕ x,h (s) ds. @PFPVA

TH

Chapter 2 e otin using iqutions @PFPSAD @PFPTAD @PFPUA nd @PFPVAD

A = n-1 i=2 Y (t i )e t i 1 t i-1 e -s ϕ x,h (s) ds + n-1 i=2 Y (t i )e -t i 1 -e -2(t i -t i-1 ) t i t i-1 e s ϕ x,h (s) ds - n-1 i=2 Y (t i )e t i 1 -e -2(t i -t i-1 ) t i t i-1 e -s ϕ x,h (s) ds + n-1 i=2 Y (t i )e -t i t i+1 0 e s ϕ x,h (s) ds + n-1 i=2 Y (t i )e t i 1 -e -2(t i+1 -t i ) t i+1 t i e -s ϕ x,h (s) ds - n-1 i=2 Y (t i )e -t i 1 -e -2(t i+1 -t i ) t i+1 t i e s ϕ x,h (s) ds - f x,h (t 2 )Y (t 1 ) 1 -e -2(t 2 -t 1 ) e -(t 2 -t 1 ) - f x,h (t n-1 )Y (t n ) 1 -e -2(tn-t n-1 ) e -(tn-t n-1 ) - n-1 i=2 Y (t i )e -t i t i 0 e s ϕ x,h (s) ds - n-1 i=2 Y (t i )e t i 1 t i e -s ϕ x,h (s) ds.
epling this expression of e in @PFPRA givesD

ĝpro n (x) = n-1 i=2 Y (t i ) t i+1 t i-1 e |t i -s| ϕ x,h (s) ds + n-2 i=2 Y (t i+1 )e -t i+1 -Y (t i )e -t i 1 -e -2(t i+1 -t i ) t i+1 t i e s ϕ x,h (s) ds - n-2 i=2 Y (t i+1 )e t i+1 -Y (t i )e t i 1 -e -2(t i+1 -t i ) t i+1 t i e -s ϕ x,h (s) ds + Y (t 2 )e -t 2 1 -e -2(t 2 -t 1 ) t 2 t 1 e s ϕ x,h (s) ds - Y (t n-1 )e -t n-1 1 -e -2(tn-t n-1 ) tn t n-1 e s ϕ x,h (s) ds - Y (t 2 )e t 2 1 -e -2(t 2 -t 1 ) t 2 t 1 e -s ϕ x,h (s) ds + Y (t n-1 )e t n-1 1 -e -2(tn-t n-1 ) tn t n-1 e -s ϕ x,h (s) ds + f x,h (t 1 )Y (t 1 ) 1 -e -2(t 2 -t 1 ) - f x,h (t 2 )Y (t 1 ) 1 -e -2(t 2 -t 1 ) e -(t 2 -t 1 ) + f x,h (t n )Y (t n ) 1 -e -2(tn-t n-1 ) - f x,h (t n-1 )Y (t n )
1 -e -2(tn-t n-1 ) e -(tn-t n-1 ) . @PFPWA xote tht iqution @PFPUA yieldsD

Y (t 1 ) 1 -e -2(t 2 -t 1 ) f x,h (t 1 ) -f x,h (t 2 )e -(t 2 -t 1 ) = Y (t 1 )e t 1 1 -e -2(t 2 -t 1 ) t 2 t 1 e -s ϕ x,h (s) ds - Y (t 1 )e -t 1 1 -e -2(t 2 -t 1 ) t 2 t 1 e s ϕ x,h (s) ds + Y (t 1 )e -t 1 t 2 t 1 e s ϕ x,h (s) ds. @PFQHA imilrlyD iqution @PFPTA yieldsD Y (t n ) 1 -e -2(tn-t n-1 ) f x,h (t n ) -f x,h (t n-1 )e -(tn-t n-1 ) = Y (t n )e -tn 1 -e -2(tn-t n-1 ) tn t n-1 e s ϕ x,h (s) ds - Y (t n )e tn 1 -e -2(tn-t n-1 ) tn t n-1 e -s ϕ x,h (s) ds + Y (t n )e tn 1 t n-1 e -s ϕ x,h (s) ds.
@PFQIA 2.7. PROOFS TI e otin using @PFQHA nd @PFQIA in @PFPWAD

ĝpro n (x) = n-1 i=2 Y (t i ) t i+1 t i-1 e |t i -s| ϕ x,h (s) ds + n-2 i=2 Y (t i+1 )e -t i+1 -Y (t i )e -t i 1 -e -2(t i+1 -t i ) t i+1 t i e s ϕ x,h (s) ds - n-2 i=2 Y (t i+1 )e t i+1 -Y (t i )e t i 1 -e -2(t i+1 -t i ) t i+1 t i e -s ϕ x,h (s) ds + Y (t 2 )e -t 2 1 -e -2(t 2 -t 1 ) t 2 t 1 e s ϕ x,h (s) ds - Y (t n-1 )e -t n-1 1 -e -2(tn-t n-1 ) tn t n-1 e s ϕ x,h (s) ds - Y (t 2 )e t 2 1 -e -2(t 2 -t 1 ) t 2 t 1 e -s ϕ x,h (s) ds + Y (t n-1 )e t n-1 1 -e -2(tn-t n-1 ) tn t n-1 e -s ϕ x,h (s) ds + Y (t 1 )e t 1 1 -e -2(t 2 -t 1 ) t 2 t 1 e -s ϕ x,h (s) ds - Y (t 1 )e -t 1 1 -e -2(t 2 -t 1 ) t 2 t 1 e s ϕ x,h (s) ds + Y (t 1 )e -t 1 t 2 t 1 e s ϕ x,h (s) ds + Y (t n )e -tn 1 -e -2(tn-t n-1 ) tn t n-1 e s ϕ x,h (s) ds - Y (t n )e tn 1 -e -2(tn-t n-1 ) tn t n-1 e -s ϕ x,h (s) ds + Y (t n )e tn 1 t n-1 e -s ϕ x,h (s) ds = n-1 i=2 Y (t i ) t i+1 t i-1 e |s-t i | ϕ x,h (s) ds + Y (t 1 ) t 2 0 e s-t 1 ϕ x,h (s) ds + Y (t n ) 1 t n-1 e tn-s ϕ x,h (s) ds - n-1 i=1 e t i+1 Y (t i+1 ) -e t i Y (t i ) 1 -e -2(t i+1 -t i ) t i+1
t i e -s ϕ x,h (s) ds

+ n-1 i=1 e -t i+1 Y (t i+1 ) -e -t i Y (t i ) 1 -e -2(t i+1 -t i ) t i+1 t i e s ϕ x,h (s) ds.
his onludes the proof of roposition PFQFPF Proof of Lemma 2.3.1.

vet (u, v) ∈ [-1, 1] 2 F e (rst onsider the tringle {-1 < u < v < 1} whih is further split into smller tringlesX D 1 = {0 < u < v < 1}, D 2 = {-1 < u < 0 < v < 1} nd D 3 = {-1 < u < v < 0}. vet b ∈]0, 1[F por (u, v) ∈ D 1 D using essumption (A)D ylor expnsion of R round (x, x) givesD R(x + bu, x + bv) = R(x, x + bv) + buR (1,0) (x, x + bv) + 1 2 b 2 u 2 R (2,0) (ε x , x + bv) = R(x, x) + bvR (0,1) (x, η x ) + buR (1,0) (x, x + bv) + 1 2 b 2 u 2 R (2,0) (ε x , x + bv), where x < ε x < x + bu < x + bv nd x < η x < x + bvF husD R(x + bu, x + bv) = R(x, x) + bvR (0,1) (x, x + ) + buR (0,1) (x, x -) + o(b).

TP

Chapter 2

xowD for (u, v) ∈ D 2 we otin in the sme wyD

R(x + bu, x + bv) = R(x, x + bv) + buR (1,0) (x, x + bv) + 1 2 b 2 u 2 R (2,0) (ε x , x + bv) = R(x, x) + bvR (0,1) (x, η x ) + buR (1,0) (x, x + bv) + 1 2 b 2 u 2 R (2,0) (ε x , x + bv), where x + bu < ε x < x < x + bv nd x < η x < x + bvF husD R(x + bu, x + bv) = R(x, x) + bvR (0,1) (x, x + ) + buR (0,1) (x, x -) + o(b). pinllyD for (u, v) ∈ D 3 we getD R(x + bu, x + bv) = R(x + bu, x) + bvR (0,1) (x + bu, x) + 1 2 b 2 v 2 R (0,2) (x + bu, η x ) = R(x, x) + ubR (1,0) (ε x , x) + bvR (0,1) (x + bu, η x ) + 1 2 b 2 v 2 R (0,2) (x + bu, η x ), where x + hu < x + bv < η x < x nd x + bu < ε x < xF husD R(x + bu, x + bv) = R(x, x) + bvR (0,1) (x, x + ) + buR (0,1) (x, x -) + o(b).
rene for v > u we hveD

R(x + bu, x + bv) = R(x, x) + B 2 R (0,1) (x, x + ) + R (0,1) (x, x -) (u + v) + B 2 R (0,1) (x, x + ) -R (0,1) (x, x -) (v -u) + o(b).
imilrlyD we otin for the tringulr

{1 > u > v > -1}D R(x + bu, x + bv) = R(x, x) + B 2 R (0,1) (x, x + ) + R (0,1) (x, x -) (u + v) + B 2 R (0,1) (x, x + ) -R (0,1) (x, x -) (u -v). husD for (u, v) ∈ [-1, 1] 2 we hveD R(x + bu, x + bv) = R(x, x) + B 2 R (0,1) (x, x + ) + R (0,1) (x, x -) (u + v) + B 2 R (0,1) (x, x + ) -R (0,1) (x, x -) |u -v|. @PFQPA
gonsider now funtion gD ounded nd integrle on [-1, 1]F he hominted gonvergene heorem yields tht R(., t) × g is n integrle funtion for every t ∈ [-1, 1]F sing @PFQPA nd puttingD

γ(x) = 1 2 R (0,1) (x, x + ) + R (0,1) (x, x -) ,
we otinD

[-1,1] 2 R(x + bu, x + bv)g(u)g(v)dudv = R(x, x) 1 -1 g(u)du 2 + 2γ(x)b 1 -1 g(u)du 1 -1 vg(v)dv - B 2 α(x) [-1,1] 2 g(u)g(v)|u -v|dudv + o(b).
@PFQQA 2.7. PROOFS TQ he left side of @PFQQA is nonEnegtive sine the utoovrine funtion R is nonEnegtive de(nite funtionF king g

(u) = u1 [-1,1] (u) we otinD 1 -1 g(u)du = 0 nd [-1,1] 2 uv|u -v|dudv = - 8 15 . husD 4 15 α(x) + o(b) ≥ 0.
king b smll enough onludes the proof of vemm PFQFIF Proof of Lemma 2.4.2.

he gret lines of this proof re sed on the work of ks nd lvisker @IWTTA @FfF vemm QFP thereAF vet x, h ∈]0, 1[ nd put g n = P Tn f x,h D it is shown y @PFIHPA in the eppendix thtD

g n (t i ) = n j=1 m x,h (t j )R(t j , t i ) for ll i = 1, • • • , n.
yn the one hndD essumption (A) yields tht g n is twie di'erentile on [0, 1] exept on T n D ut it hs left nd right derivtivesF husD for every i = 1, . . . , n we hveD

g n (t - i ) = n j=1 m x,h (t j )R (0,1) (t j , t - i ) nd g n (t + i ) = n j=1 m x,h (t j )R (0,1) (t j , t + i ).
ine for j = iD R (0,1) (t j , t - i ) = R (0,1) (t j , t + i ) then essumption (B) yieldsD

g n (t - i ) -g n (t + i ) = α(t i )m x,h (t i ).
@PFQRA yn the other hndD essumption (A) yields tht f x,h @s de(ned y @PFPAA is twie di'erentile on ]0, 1[D thus for i = 1, . . . , n -1D ylor expnsion of f x,h -g n round t i givesD

f x,h (t i+1 ) -g n (t i+1 ) = (f x,h (t i ) -g n (t i )) + d i (f x,h (t i ) -g n (t + i )) + 1 2 d 2 i (f x,h (σ i ) -g n (σ i )),
where

d i = t i+1 -t i nd σ i ∈]t i , t i+1 [F ell thtD for ll i = 1, . . . , nD f x,h (t i ) = g n (t i ) @see iqution @PFTAAF husD f x,h (t i ) -g n (t + i ) = - 1 2 d i (f x,h (σ i ) -g n (σ i )), @PFQSA
imilrlyD for i = 2, . . . , nD we hveD

f x,h (t i ) -g n (t - i ) = 1 2 d i-1 (f x,h (θ i ) -g n (θ i )), @PFQTA
for some θ i ∈]t i-1 , t i [F e otin sutrting @PFQTA from @PFQTA nd using @PFQRA for i = 2, . . . , n-1D

α(t i )m x,h (t i ) = - 1 2 d i (f x,h (σ i ) -g n (σ i )) - 1 2 d i-1 (f x,h (θ i ) -g n (θ i )). @PFQUA

TR

Chapter 2

e shll now ontrol the lst expressionF yn the one hnd we hveD

f x,h (t) = t 0 R (0,1) (s, t + )ϕ x,h (s) ds + 1 t R (0,1) (s, t -)ϕ x,h (s) ds, @PFQVA ndD f x,h (t) = (R (0,1) (t, t + ) -R (0,1) (t, t -))ϕ x,h (t) + 1 0 R (0,2) (s, t + )ϕ x,h (s) ds -α(t)ϕ x,h (t) + 1 0 R (0,2) (s, t + )ϕ x,h (s) ds. @PFQWA
yn the other hnd we knowD using @pQA in the eppendixD tht every funtion in the ur@RAD noted y

F(ε)D is ontinuousD hene essumption (C) implies tht R (0,2) (•, t + ) is ontinuous funtion on [0, 1] for every (xed t ∈ [0, 1]F husD R (0,2) (t, t + ) = lim s↓t R (0,2) (s, t + ) = lim s↓t R (0,2) (s, t -) = R (0,2) (t, t -),
from whih we get tht R (0,2) (t, t) existsF rene for i = 1, . . . , n we hveD

g n (t - i ) = g n (t + i ) = n j=1 m x,h (t j )R (0,2) (t j , t i ).
@PFRHA sn dditionD it is shown y @pRA in the eppendix tht for every t ∈ [0, 1]D

f x,h (t) -g n (t) = -α(t)ϕ x,h (t) + R (0,2) (•, t), f x,h -g n , @PFRIA
where •, • is the inner produt on F(ε)F snjeting @PFRIA in @PFQUA we otinD

α(t i )m x,h (t i ) = 1 2 d i α(σ i )ϕ x,h (σ i ) + 1 2 d i-1 α(θ i )ϕ x,h (θ i ) - 1 2 d i R (0,2) (•, σ i ), f x,h -g n - 1 2 d i-1 R (0,2) (•, θ i ), f x,h -g n .
sing essumption (B) we otin for i = 2, . . . , n -1D

m x,h (t i ) = 1 2 (d i + d i-1 )ϕ x,h (t i ) + 1 2α(t i ) d i α(σ i )ϕ x,h (σ i ) -α(t i )ϕ x,h (t i ) + 1 2α(t i ) d i-1 α(θ i )ϕ x,h (θ i ) -α(t i )ϕ x,h (t i ) - 1 2α(t i ) d i R (0,2) (•, σ i ), f x,h -g n - 1 2α(t i ) d i-1 R (0,2) (•, θ i ), f x,h -g n ∆ = 1 2 (d i + d i-1 )ϕ x,h (t i ) + A (1) i + A (2) i -A (3) i -A (4)
i , @PFRPA sing the guhyEhwrtz inequlityD essumption (C) nd iqution @PFSSA @in the proof of roposition PFRFQ elowA we otinD

|A (3) i + A (4) i | ≤ sup 0≤t≤1 1 2α(t) ||R (0,2) (., t)|| √ C √ h sup 0≤j≤n d 2 j ∆ = β n,h , @PFRQA 2.7. PROOFS

TS

where C is positive onstnt de(ned in roposition PFRFQ elowF

ell tht ϕ x,h is of support [x -h, x + h]D thus for t i suh tht [t i-1 , t i+1 ]∩]x -h, x + h[= ∅D ϕ x,h (t) = 0 so tht A (1) i = 0 nd A (2) i = 0F por t i suh tht [t i-1 , t i+1 ]∩]x -h, x + h[ = ∅D letD α n,h = sup 0≤i≤n sup t i ≤s,t≤t i+1 1 2α(t) d i |α(s)ϕ x,h (s) -α(t)ϕ x,h (t)|.
@PFRRA e otin using @PFRQA nd @PFRRA together with @PFRPA for i = 2, . . . , n -1D

m x,h (t i ) = 1 2 ϕ x,h (t i )(t i+1 -t i-1 ) + O α n,h + β n,h if [t i-1 , t i+1 ]∩]x -h, x + h[ = ∅ O β n,h otherwise.
efter hving otined m x,h (t i ) for i = 2, . . . , n -1D we re now le to otin m x,h (t 1 ) nd m x,h (t n )F e hve for i = 1, . . . , nD

R(t 1 , t i )m x,h (t 1 ) + R(t n , t i )m x,h (t n ) = f x,h (t i ) - n-1 j=2 m x,h (t j )R(t j , t i ). @PFRSA ell tht N Tn = Card I x,h = Card {i = 1, • • • , n : [t i-1 , t i+1 ]∩]x -h, x + h[ = ∅} nd tht t x,i re the points of T n for whih i ∈ I x,h F e hveD n-1 j=2 m x,h (t j )R(t j , t i ) = N Tn j=1 m x,h (t x,j )R(t x,j , t i ) + n-1 j=2 1 {j / ∈I x,h } m x,h (t j )R(t j , t i ).
yn the one hndD we hve using @PFRPA @where A x,j stnds for A j with t j repled y t x,j AD

n-1 j=2 m x,h (t j )R(t j , t i ) = 1 2 N Tn j=1 (d x,j + d x,j-1 )ϕ x,h (t x,j )R(t x,j , t i ) + N Tn j=1 (A 1 x,j + A 2 x,j -A 3 x,j -A 4 x,j )R(t x,j , t i ) - n-1 j=2 1 {j / ∈I x,h } (A 3 j + A 4 j )R(t j , t i ) = 1 2 N Tn j=1 (d x,j + d x,j-1 )ϕ x,h (t x,j )R(t x,j , t i ) + N Tn j=1 (A 1 x,j + A 2 x,j )R(t x,j , t i ) - n j=1 (A 3 j + A 4 j )R(t j , t i ). (2.46)
yn the other hndD

f x,h (t i ) = 1 0 R(s, t i )ϕ x,h (s) ds = x+h x-h R(s, t i )ϕ x,h (s) ds = 1 2 N Tn j=1 t x,j+1 t x,j-1 R(s, t i )ϕ x,h (s) ds = 1 2 N Tn j=1 (d x,j + d x,j-1 )R(t x,j , t i )ϕ x,h (t j ) + 1 2 N Tn j=1 t x,j+1 t x,j-1 (R(s, t i )ϕ x,h (s) -R(t x,j , t i )ϕ x,h (t x,j )) ds.
@PFRUA snserting @PFRTA nd @PFRUA in @PFRSA we otin for i = 1, . . . , nD

R(t 1 , t i )m x,h (t 1 ) + R(t n , t i )m x,h (t n ) = 1 2 N Tn j=1 t x,j+1 t x,j-1 (R(s, t i )ϕ x,h (s) -R(t x,j , t i )ϕ x,h (t x,j )) ds - N Tn j=1 (A 1 x,j + A 2 x,j )R(t x,j , t i ) + n j=1 (A 3 j + A 4 j )R(t j , t i ) ∆ = Φ x,h (t i ).

TT

Chapter 2 e then otin the following liner systemD

     R(t 1 , t 1 )m x,h (t 1 ) + R(t n , t 1 )m x,h (t 1 ) = Φ x,h (t 1 ). R(t 1 , t n )m x,h (t 1 ) + R(t n , t n )m x,h (t n ) = Φ x,h (t n ).
@PFRVA olving @PFRVA for m x,h (t 1 ) nd m x,h (t n ) we otinD

m x,h (t 1 ) = R(t n , t n )Φ x,h (t 1 ) -R(t 1 , t n )Φ x,h (t n ) R(t 1 , t 1 )R(t n , t n ) -R(t 1 , t n ) 2 . m x,h (t n ) = R(t 1 , t 1 )Φ x,h (t n ) -R(t 1 , t n )Φ x,h (t 1 ) R(t 1 , t 1 )R(t n , t n ) -R(t 1 , t n ) 2 .
pinllyD simple lultions yieldD

m x,h (t 1 ) = O(N Tn α n,h + nβ n,h nd m x,h (t n ) = O(N Tn α n,h + nβ n,h .
his ompletes the proof of vemm PFRFPF Proof of Proposition 2.4.1.

ell tht N Tn = Card I x,h = Card {i = 1, • • • , n : [t i-1 , t i+1 ]∩]x -h, x + h[ = ∅} nd denote y t x,i the points of T n for whih i ∈ I x,h D tht is T n ∩]x -h, x + h[= {t x,2 , • • • , t x,N Tn -1 }F ine E(Y (t i )) = g(t i ) thenD E(ĝ pro n (x)) = n j=1 m x,h (t j )g(t j ) = N Tn i=1 m x,h (t x,i )g(t x,i ) + n-1 j=2 1 {i / ∈I x,h } m x,h (t j )g(t j ) + m x,h (t 1 )g(t 1 ) + m x,h (t n )g(t n ).
sing the symptoti pproximtion of m x,h |Tn given in vemm PFRFP we otinD

E(ĝ pro n (x)) = 1 2 N Tn i=1 (t x,i+1 -t x,i-1 )ϕ x,h (t x,i )g(t x,i ) + O N Tn α n,h + nβ n,h , @PFRWA por x ∈ [0, 1] letD I h (x) = x+h x-h ϕ x,h (t)g(t) dt = 1 2 N Tn i=1 t x,i+1 t x,i-1 ϕ x,h (t)g(t) dt, nd writeD E(ĝ pro n (x)) = E(ĝ pro n (x)) -I h (x) + I h (x) = ∆ x,h + I h (x) + O N Tn α n,h + nβ n,h , @PFSHA whereD ∆ x,h = 1 2 N Tn i=1 t x,i+1 t x,i-1 ϕ x,h (t x,i )g(t x,i ) -ϕ x,h (t)g(t) dt. 2.7. PROOFS TU e (rst ontrol ∆ x,h F e hveD ∆ x,h = 1 2 N Tn i=1 ϕ x,h (t x,i ) t x,i+1 t x,i-1 (g(t x,i ) -g(t)) dt + 1 2 N Tn i=1 t x,i+1 t x,i-1 g(t)(ϕ x,h (t x,i ) -ϕ x,h (t)) dt. ine ϕ x,h is in C 1 nd g is in C 2 then ylor expnsions of ϕ x,h nd g giveD g(t) = g(t x,i ) + (t -t x,i )g (t x,i ) + 1 2 (t -t x,i ) 2 g (θ x,i ), ndD ϕ x,h (t) = ϕ x,h (t x,i ) + (t -t x,i )ϕ x,h (η x,i ),
for some θ x,i nd η x,i etween t nd t x,i F husD

∆ x,h = - 1 2 N Tn i=1 ϕ x,h (t x,i )g (t x,i ) t x,i+1 t x,i-1 (t -t x,i ) dt - 1 4 N Tn i=1 ϕ x,h (t x,i ) t x,i+1 t x,i-1 g (θ x,i )(t -t x,i ) 2 dt - 1 2 N Tn i=1 g(t x,i ) t x,i+1 t x,i-1 ϕ x,h (η x,i )(t -t x,i ) dt - 1 2 N Tn i=1 g (t x,i ) t x,i+1 t x,i-1 ϕ x,h (η x,i )(t -t x,i ) 2 dt - 1 4 N Tn i=1 t x,i+1 t x,i-1 g (θ x,i )ϕ x,h (η x,i )(t -t x,i ) 3 dt.
ell tht g nd g re oth ounded nd thtD

sup 0≤t≤1 |ϕ x,h (t)| < c h nd sup 0≤t≤1 |ϕ x,h (t)| < c h 2 , @PFSIA
for pproprite positive onstnts c nd c F sing this we otinD

1 4 N Tn i=1 ϕ x,h (t x,i ) t x,i+1 t x,i-1 g (θ x,i )(t -t x,i ) 2 dt = O N Tn h sup 0≤j≤1 d 3 j,n 1 2 N Tn i=1 g (t x,i ) t x,i+1 t x,i-1 ϕ x,h (η x,i )(t -t x,i ) 2 dt = O N Tn h 2 sup 0≤j≤1 d 3 j,n 1 4 N Tn i=1 t x,i+1 t x,i-1 g (θ x,i )ϕ x,h (η x,i )(t -t x,i ) 2 dt = O N Tn h 2 sup 0≤j≤1 d 3 j,n . husD ∆ x,h = - 1 2 N Tn i=1 ϕ x,h (t x,i )g (t x,i ) t x,i+1 t x,i-1 (t -t x,i )dt - 1 2 N Tn i=1 g(t x,i )ϕ x,h (t x,i ) t x,i+1 t x,i-1 (t -t x,i ) dt - 1 2 N Tn i=1 g(t x,i ) t x,i+1 t x,i-1 (t -t x,i ) ϕ x,h (η x,i ) -ϕ x,h (t x,i ) dt + O N Tn h 2 sup 0≤j≤1 d 3 j,n . TV Chapter 2 ine ϕ x,h is vipshitz thenD N Tn i=1 g(t x,i ) t x,i+1 t x,i-1 (t -t x,i ) ϕ x,h (η x,i ) -ϕ x,h (t x,i ) dt = O N Tn h 3 sup 0≤j≤1 d 3 j,n . husD ∆ x,h = - 1 2 N Tn i=1 ϕ x,h (t x,i )g (t x,i ) t x,i+1 t x,i-1 (t -t x,i )dt - 1 2 N Tn i=1 g(t x,i )ϕ x,h (t x,i ) t x,i+1 t x,i-1 (t -t x,i ) dt + O N Tn h 3 sup 0≤j≤1 d 3 j,n .
fsi integrtion givesD

∆ x,h = - 1 4 N Tn i=1 ϕ x,h (t x,i )g (t x,i )(d 2 x,i -d 2 x,i-1 ) - 1 4 N Tn i=1 g(t x,i )ϕ x,h (t x,i )(d 2 x,i -d 2 x,i-1 ) + O N Tn h 3 sup 0≤j≤1 d 3 j,n .
e shll show thtD

A ∆ = N Tn i=1 ϕ x,h (t x,i )g (t x,i )(d 2 x,i -d 2 x,i-1 ) = O N Tn h 2 sup 0≤j≤1 d 3 j,n , B ∆ = N Tn i=1 g(t x,i )ϕ x,h (t x,i )(d 2 x,i -d 2 x,i-1 ) = O N Tn h 3 sup 0≤j≤1 d 3 j,n .
trting with the term AF ell thtD sine ϕ is of support

[x -h, x + h] nd t x,1 , t x,N Tn -1 / ∈ ]x -h, x + h[D then ϕ x,h (t x,N Tn ) = ϕ x,h (t x,1 ) = 0 thusD A = N Tn -1 i=2 ϕ x,h (t x,i )g (t x,i )d 2 x,i - N Tn -2 i=1 ϕ x,h (t x,i+1 )g (t x,i+1 )d 2 x,i = N Tn -2 i=2 ϕ x,h (t x,i )g (t x,i ) -ϕ x,h (t x,i+1 )g (t x,i+1 ) d 2 x,i + ϕ x,h (t x,N Tn -1 )g (t x,N Tn -1 )d 2 x,N Tn -1 -ϕ x,h (t x,2 )g (t x,2 )d 2 x,1 ∆ = A 1 + A 2 .
yn the one hndD ylor expnsions of ϕ x,h round t x,N Tn nd t x,1 yieldD

ϕ x,h (t x,N Tn -1 ) = (t x,N Tn -1 -t x,N Tn )ϕ x,h (γ x,N Tn ), ϕ x,h (t x,2 ) = (t x,2 -t x,1 )ϕ x,h (γ x,1 ),
for some γ x,N Tn ∈]t x,N Tn -1 , t x,N Tn [ nd some γ x,1 ∈]t x,1 , t x,2 [F sing @PFSIA nd the ft tht g is ounded we otinD

A 2 = O 1 h 2 sup 0≤j≤1 d 3 j,n .

PROOFS

TW yn the other hnd we hveD

A 1 = N Tn -2 i=2 ϕ x,h (t x,i )g (t x,i ) -ϕ x,h (t x,i+1 )g (t x,i+1 ) d 2 x,i = N Tn -2 i=2 ϕ x,h (t x,i ) g (t x,i ) -g (t x,i+1 ) d 2 x,i + N Tn -2 i=2 g (t x,i+1 ) ϕ x,h (t x,i ) -ϕ x,h (t x,i+1 ) d 2 x,i .
ine ϕ x,h is in C 1 nd g is in C 2 then using @PFSIAD we otinD

A 1 = O N Tn h 2 sup 0≤j≤1 d 3 j,n .
sn similr wy nd from essumption (D)D we otinD

B = O N Tn h 3 sup 0≤j≤1 d 3 j,n . reneD ∆ x,h = O N Tn h 3 sup 0≤j≤1 d 3 j,n .
hus using @PFSHAD

E(ĝ pro n (x)) = I h (x) + O(N Tn α n,h + nβ n,h ) + O N Tn h 3 sup 0≤j≤1 d 3 j,n .
he ontrol of I h (x) is lssil nd it n ee seen from qsser nd wüller @IWVRA thtD

I h (x) = g(x) + 1 2 h 2 g (x) 1 -1 t 2 K(t) dt + o(h 2 ). @PFSPA pinllyD E(ĝ pro n (x)) -g(x) = 1 2 h 2 g (x) 1 -1 t 2 K(t) dt + o(h 2 ) + O N Tn h 3 sup 0≤j≤1 d 3 j,n + N Tn α n,h + nβ n,h .
his onludes the proof of roposition PFRFIF Proof of Proposition 2.4.2.

vet t 0 = 0, t n+1 = 1 nd set Y (t 0 ) = 0 nd Y (t n+1 ) = Y (t n )F ell thtD ĝpro n (x) = n+1 i=1 Y (t i ) t i t i-1 ϕ x,h (s)ds + n i=0 Y (t i+1 ) -Y (t i ) t i+1 -t i t i+1 t i (s -t i+1 )ϕ x,h (s)ds. ine E (Y (t i )) = g(t i ) thenD E (ĝ pro n (x)) = n+1 i=1 g(t i ) t i t i-1 ϕ x,h (s)ds + n i=0 g(t i+1 ) -g(t i ) t i+1 -t i t i+1 t i (s -t i+1 )ϕ x,h (s)ds. UH Chapter 2 ell tht N Tn = Card I x,h = {i = 1, • • • , n : [t i-1,n , t i+1,n ]∩]x -h, x + h[ = ∅} nd denote y t x,i
the points of T n for whih i ∈ I x,h F sing the support of ϕ x,h we otinD

E (ĝ pro n (x)) = N Tn i=1 g(t x,i ) t x,i t x,i-1 ϕ x,h (s)ds + N Tn i=1 g(t x,i+1 ) -g(t x,i ) t x,i+1 -t x,i t x,i+1 t x,i (s -t x,i+1 )ϕ x,h (s)ds. vet d x,i = t x,i+1 -t x,i F ine g is in C 2 nd ϕ x,h is in C 1 then ylor expnsions of g round t x,i nd of ϕ x,h round t x,i+1 yieldD g(t x,i+1 ) = g(t x,i ) + d x,i g (t x,i ) + 1 2 d 2 x,i g (θ x,i ), ϕ x,h (s) = ϕ x,h (t x,i+1 ) + (s -t x,i+1 )ϕ x,h (s i ). for some θ x,i ∈]t x,i , t x,i+1 [ nd some s i ∈]s, t x,i+1 [F ell thtD using the support of ϕD ϕ x,h (t x,1 ) = ϕ x,h (t x,N Tn ) = 0 thusD E (ĝ pro n (x)) = N Tn i=1 g(t x,i ) t x,i t x,i-1 ϕ x,h (s)ds + N Tn -2 i=1 g (t x,i )ϕ x,h (t x,i+1 ) t x,i+1 t x,i (s -t x,i+1 )ds + N Tn i=1 g (t x,i ) t x,i+1 t x,i (s -t x,i+1 ) 2 ϕ x,h (s i )ds + 1 2 N Tn i=1 ϕ x,h (t x,i+1 )g (θ x,i )d x,i t x,i+1 t x,i (s -t x,i+1 )ds + 1 2 N Tn i=1 g (θ x,i )d x,i t x,i+1 t x,i (s -t x,i+1 ) 2 ϕ x,h (s i )ds.
ell tht g nd g re oundedD vemm PFRFI yields N Tn = O(nh) nd d x,i = O( 1 n ) nd using @PFSIA we otinD

N Tn i=1 g (t x,i ) t x,i+1 t x,i (s -t x,i+1 ) 2 ϕ x,h (s i )ds = O 1 n 2 h . 1 2 N Tn i=1 ϕ x,h (t x,i+1 )g (θ x,i )d x,i t x,i+1 t x,i (s -t x,i+1 )ds = O 1 n 2 . 1 2 N Tn i=1 g (θ x,i )d x,i t x,i+1 t x,i (s -t x,i+1 ) 2 ϕ x,h (s i )ds = O 1 n 3 h .
st follows tht y simple integrtionD

E (ĝ pro n (x)) = N Tn i=1 g(t x,i ) t x,i t x,i-1 ϕ x,h (s)ds - 1 2 N Tn -2 i=1 g (t x,i )ϕ x,h (t x,i+1 )d 2 x,i + O 1 n 2 h = N Tn i=1 t x,i t x,i-1 ϕ x,h (s)g(s) ds + N Tn i=1 t x,i t x,i-1 ϕ x,h (s)(g(t x,i ) -g(s)) ds - 1 2 N Tn -2 i=1 g (t x,i )ϕ x,h (t x,i+1 )d 2 x,i + O 1 n 2 h .

PROOFS

UI yn the one hndD we hveD

N Tn i=1 t x,i t x,i-1 ϕ x,h (s)g(s) ds = x+h x-h ϕ x,h (s)g(s) ds.
yn the other hndD ylor expnsion of g nd ϕ x,h rround t x,i yieldD

g(t x,i ) = g(s) + (t x,i -s)g (t x,i ) - 1 2 (t x,i -s) 2 g (s i ), ϕ x,h (s) = ϕ x,h (t x,i ) + (s -t x,i )ϕ x,h (s i ).
for some

s i nd s i in ]s, t x,i [F husD E (ĝ pro n (x)) = x+h x-h ϕ x,h (s)g(s) ds + N Tn -1 i=2 g (t x,i )ϕ x,h (t x,i ) t x,i t x,i-1 (t x,i -s) ds - N Tn i=1 g (t x,i ) t x,i t x,i-1 (t x,i -s) 2 ϕ x,h (s i ) ds - 1 2 N Tn i=1 ϕ x,h (t x,i ) t x,i t x,i-1 g (s i )(t x,i -s) 2 ds + 1 2 N Tn i=1 t x,i t x,i-1 g (s i )ϕ x,h (s i )(t x,i -s) 3 ds - 1 2 N Tn -2 i=1 g (t x,i )ϕ x,h (t x,i+1 )d 2 x,i + O 1 n 2 h .
sing the oundedness of g nd g in ddition to vemm PFRFI nd iqution @PFSIAD we otinD

N Tn i=1 g (t x,i ) t x,i t x,i-1 (t x,i -s) 2 ϕ x,h (s i ) ds = O 1 n 2 h . 1 2 N Tn i=1 ϕ x,h (t x,i ) t x,i t x,i-1 g (s i )(t x,i -s) 2 ds = O 1 n 2 . 1 2 N Tn i=1 t x,i t x,i-1 g (s i )ϕ x,h (s i )(t x,i -s) 3 ds = O 1 n 3 h . husD E (ĝ pro n (x)) = x+h x-h ϕ x,h (s)g(s) ds + 1 2 N Tn -2 i=2 g (t x,i )ϕ x,h (t x,i )d 2 x,i-1 - 1 2 N Tn -2 i=1 g (t x,i )ϕ x,h (t x,i+1 )d 2 x,i + O 1 n 2 h = x+h x-h ϕ x,h (s)g(s) ds + 1 2 N Tn -2 i=1 g (t x,i+1 ) -g (t x,i ) ϕ x,h (t x,i+1 )d 2 x,i + O 1 n 2 h .
ine g is vipshitzD then we hveD

E (ĝ pro n (x)) = x+h x-h ϕ x,h (s) g(s) ds + O 1 n 2 h . @PFSQA UP Chapter 2
pinllyD from @PFSPA we otinD

E (ĝ pro n (x)) -g(x) = 1 2 h 2 g (x) 1 -1 t 2 K(t)dt + o(h 2 ) + O 1 n 2 h .
his onludes the proof of roposition PFRFPF Proof of Proposition 2.4.3.

he gret lines of this proof re sed on ks nd lvisker @IWTTAF prom the de(nition of the orthogonl projetion @see etion PFPA nd using the ythgore theorem we otinD

m σ 2 x,h m -Varg pro n (x) = ||f x,h || 2 -||P |Tn f x,h || 2 = ||f x,h -P |Tn f x,h || 2 , @PFSRA
where P |Tn f x,h is the orthogonl projetion of f x,h on the suspe of F(ε) spnned y {R(•, t i ), t i ∈ T n }D denoted here y V Tn F e shll then prove thtD

||f x,h -P |Tn f x,h || 2 ≤ C h sup 0≤j≤n d 2 j,n . @PFSSA ell tht N Tn = Card I x,h = Card I x,h = {i = 1, • • • , n : [t i-1,n , t i+1,n ]∩]x -h, x + h[ = ∅} nd denote y t x,i the points of T n for whih i ∈ I x,h F vet g n := g n,x = n i=1 γ x,i R(•, t x,i ) with γ x,i = 0 for every i / ∈ I x,h F st is ler tht g n ∈ V
Tn nd thus from the de(nition of the orthogonl projetion we hveD

||f x,h -P |Tn f x,h || 2 ≤ ||f x,h -g n || 2 .
xow using @pIA in the eppendix nd the support of ϕ x,h we otinD

||f x,h -g n || 2 = 1 0 (f x,h (t) -g n (t))ϕ x,h (t) dt - n i=1 (f x,h (t i ) -g n (t i ))γ x,i = x+h x-h (f x,h (t) -g n (t))ϕ x,h (t) dt - N Tn i=1 (f x,h (t x,i ) -g n (t x,i ))γ x,i @PFSTA
sn wht followsD we distinguish etween three ses ording to the lotion of t x,1 nd t

x,N Tn in the intervl [x -h, x + h]F First case. uppose (rst tht t x,1 = x -h nd t x,N Tn = x + h nd tkeD γ x,i = t x,i+1 t x,i ϕ x,h (t) dt for i = 1, . . . , N Tn -1. @PFSUA
we hve in this seD

||f x,h -g n || 2 = N Tn i=1 t x,i+1 t x,i (f x,h (t) -g n (t)) -(f x,h (t x,i ) -g n (t x,i )) ϕ x,h (t) dt. @PFSVA essumption (A) yields tht f x,h is twie di'erentile on [0, 1]D while g n is twie di'erentile everywhere exept on T n D ut it hs left nd right derivtivesF ylor expnsion of f x,h -g n round t x,i for i = 1, • • • , N Tn -1 nd t ∈]t x,i , t x,i+1 [ givesD f x,h (t) -g n (t) = (f x,h (t x,i ) -g n (t x,i )) + (t -t x,i )(f x,h (t x,i ) -g n (t + x,i )) + 1 2 (t -t x,i ) 2 (f x,h (θ x,t ) -g n (θ + x,t )), @PFSWA 2.7. PROOFS

UQ

for some θ x,t ∈]t x,i , t[F yn the one hndD we hveD

g n (t + x,i ) = N Tn -1 j=1 R (0,1) (t x,j , t + x,i )γ x,j .
@PFTHA yn the other hndD using @PFQVA we otinD

f x,h (t x,i ) = x+h x-h R (0,1) (s, t + x,i )ϕ x,h (s) ds = N Tn -1 j=1 t x,j+1 t x,j R (0,1) (s, t + x,i )ϕ x,h (s) ds = N Tn -1 j=1 j =i t x,j+1 t x,j R (0,1) (s, t + x,i )ϕ x,h (s) ds + t x,i+1 t x,i R (0,1) (s, t + x,i )ϕ x,h (s) ds. @PFTIA hen j = i we hveD tx,j+1 tx,j R (0,1) (s, t + x,i )ϕ x,h (s) ds = R (0,1) (t x,j , t x,i )γ x,j + tx,j+1 tx,j (s -t x,j )R (1,1) (δ s,j , t x,i )ϕ x,h (s) ds, (2.62)
for some δ s,j ∈]t x,j , s[D while for j = i we hveD

t x,i+1 t x,i R (0,1) (s, t + x,i )ϕ x,h (s) ds = t x,i+1 t x,i R (0,1) (s, t - x,i )ϕ x,h (s) ds = R (0,1) (t x,i , t - x,i )γ x,i + t x,i+1 t x,i (s -t x,i )R (1,1) (δ + s,i , t - x,i )ϕ x,h (
s) ds. @PFTQA golleting @PFTHAD @PFTIAD @PFTPA nd @PFTQA we otinD

f x,h (t x,i ) -g n (t + x,i ) = N Tn -1 j=1 j =i R (0,1) (t x,j , t x,i )γ x,j + N Tn -1 j=1 j =i t x,j+1 t x,j (s -t x,j )R (1,1) (δ s,j , t x,i )ϕ x,h (s) ds + R (0,1) (t x,i , t - x,i )γ x,i + t x,i+1 t x,i R (1,1) (δ + s,i , t - x,i )ϕ x,h (s) ds - N Tn -1 j=1 R (0,1) (t x,j , t + x,i )γ x,j = α(t x,i )γ x,i + N Tn -1 j=1 t x,j+1 t x,j (s -t x,j )R (1,1) (δ + s,j , t - x,i )ϕ x,h (s) ds.
st is esy to see thtD

|f x,h (t x,i ) -g n (t + x,i )| ≤ α 1 γ x,i + K ∞ h R 1 N Tn -1 j=1 t x,j+1 t x,j (s -t x,j ) ds ≤ K ∞ h α 1 d x,i + K ∞ 2h R 1 N Tn -1 j=1 d 2 x,j .
@PFTRA e dedue from @PFQWA tht for ll θ x,t ∈]t x,i , t x,i+1 [ we hveD

|f x,h (θ x,t )| ≤ K ∞ h α 1 + K ∞ h R 2 × 2h = K ∞ h α 1 + 2K ∞ R 2 .
sn dditionD for θ x,t ∈]t x,i , t x,i+1 [ we hveD

|g n (θ + x,t )| = N Tn -1 j=1 R (0,2) (t x,j , θ + x,t )γ x,j ≤ K ∞ h R 2 N Tn -1 j=1 d x,j = K ∞ h R 2 × 2h = 2K ∞ R 2 , husD |f x,h (θ x,t ) -g n (θ + x,t )| ≤ K ∞ h α 1 + 4K ∞ R 2 . @PFTSA iqutions @PFSWAD @PFTRA nd @PFTSA yield tht for i = 1, • • • , N Tn -1, tx,i+1 tx,i (f x,h (t) -g n (t)) -(f x,h (t x,i ) -g n (t x,i )) ϕ x,h (t) dt ≤ tx,i+1 tx,i (t -t x,i )|f x,h (t x,i ) -g n (t + x,i )||ϕ x,h (t)| dt + 1 2 tx,i+1 tx,i (t -t x,i ) 2 |f x,h (θ x,t ) -g n (θ + x,t )||ϕ x,h (t)| dt ≤ K ∞ h α 1 d x,i + K ∞ 2h R 1 N Tn -1 j=1 d 2 x,j tx,i+1 tx,i (t -t x,i )|ϕ x,h (t)| dt + 1 2 K ∞ h α 1 + 4K ∞ R 2 tx,i+1 tx,i (t -t x,i ) 2 |ϕ x,h (t)| dt ≤ K ∞ h α 1 d x,i + K ∞ 2h R 1 N Tn -1 j=1 d 2 x,j K ∞ 2h d 2 x,i + 1 2 K ∞ h α 1 + 4K ∞ R 2 K ∞ 3h d 3 x,i ≤ K 2 ∞ 4h 2 R 1 d 2 x,i N Tn -1 j=1 d 2 x,j + 2K 2 ∞ 3h α 1 h + R 2 d 3 x,i .
(2.66) snjeting this inequlity in @PFSVA yieldsD sing this we otinD

||f x,h -P |Tn f x,h || 2 ≤ K 2 ∞ 4h 2 R 1 N Tn -1 i=1 d 2 x,i 2 + 2K 2 ∞ 3h α 1 h + R 2 N Tn -1 i=1 d 3 x,i ≤ K 2 ∞ 4h 2 R 1 sup 1≤i≤n d 2 i,n N Tn -1 i=1 d x,i 2 + 2K 2 ∞ 3h α 1 h + R 2 sup 1≤i≤n d 2 i,n N Tn -1 i=1 d x,i . ine N Tn -1 i=1 d x,i = 2h thenD ||f x,h -P |Tn f x,h || 2 ≤ 4 3h α 1 + R 1 + 4 3 R 2 K 2 ∞ sup 1≤i≤n d 2 i,n pinllyD sine h < 1 thenD ||f x,h -P |Tn f x,h || 2 ≤ 4 3 α 1 + R 1 + 4 3 R 2 K 2 ∞ 1 h sup 1≤i≤n d 2 i,
||f x,h -g n || 2 = t x,2
x-h

(f x,h (t) -g n (t)) -(f x,h (t x,1 ) -g n (t x,1 )) ϕ x,h (t) dt + N Tn i=2 t x,i+1 t x,i (f x,h (t) -g n (t)) -(f x,h (t x,i ) -g n (t x,i )) ϕ x,h (t) dt + x+h t x,N Tn (f x,h (t) -g n (t)) -(f x,h (t x,N Tn ) -g n (t x,N Tn )) ϕ x,h (t) dt.
@PFTUA e (rst ontrol the (rst term of @PFTUAF vetD

A (1)
x,h = t x,2

x-h

(f x,h (t) -g n (t)) -(f x,h (t x,1 ) -g n (t x,1 )) ϕ x,h (t) dt. por t ∈]x -h, t x,2 [ we hveD f x,h (t) -g n (t) = (f x,h (t x,1 ) -g n (t x,1 )) + (t -t x,1 )(f x,h (t x,1 ) -g n (t + x,1 )) + 1 2 (t -t x,1 ) 2 (f x,h (θ x,1 ) -g n (θ + x,1 )), @PFTVA for some θ x,1 ∈]x -h, t[F iqution @PFQVA yieldsD f x,h (t x,1 ) = x+h x-h R (0,1) (s, t + x,1 )ϕ x,h (s) ds = N Tn -1 j=1 t x,j+1 t x,j R (0,1) (s, t + x,1 )ϕ x,h (s) ds = t x,2 x-h R (0,1) (s, t - x,1 )ϕ x,h (s) ds + N Tn -1 j=2 t x,j+1 t x,j R (0,1) (s, t + x,1 )ϕ x,h (s) ds = R (0,1) (t x,1 , t - x,1 )γ x,1 + t x,2 x-h (s -t x,1 )R (1,1) (δ + s,1 , t - x,1 )ϕ x,h (s) ds + N Tn -1 j=2 R (0,1) (t x,j , t x,1 )γ x,j + N Tn -1 j=2 t x,j+1 t x,j (s -t x,j )R (1,1) (δ s,j , t + x,1 )ϕ x,h (s) ds. @PFTWA ell thtD g n (t + x,1 ) = R (0,1) (t x,1 , t + x,1 )γ x,1 + N Tn -1 j=2
R (0,1) (t x,j , t x,1 )γ x,j . @PFUHA iqutions @PFTWA nd @PFUHA giveD

f x,h (t x,1 ) -g n (t + x,1 ) = α(t x,1 )γ x,1 + N Tn -1 j=2 t x,j+1 t x,j (s -t x,j )R (1,1) (δ s,j , t + x,1 )ϕ x,h (s) ds + t x,2 x-h (s -t x,1 )R (1,1) (δ + s,1 , t - x,1 )ϕ x,h (s) ds. UT Chapter 2 xote tht t x,2 -(x -h) ≤ sup 1≤i≤n d i,n F e otinD |f x,h (t x,1 ) -g n (t - x,1 )| ≤ K ∞ h α 1 sup 1≤i≤n d i,n + K ∞ 2h R 1 N Tn -1 j=2 d 2 x,j + K ∞ 2h R 1 sup 1≤i≤n d 2 i,n ≤ K ∞ h α 1 sup 1≤i≤n d i,n + K ∞ R 1 sup 1≤i≤n d i,n + K ∞ 2h R 1 sup 1≤i≤n d 2 i,n ≤ K ∞ α 1 h + 3 2 R 1 sup 1≤i≤n d i,n @PFUIA
fy @PFTSA we hveD

|f x,h (θ x,t ) -g n (θ - x,t )| ≤ K ∞ h α 1 + 4K ∞ R 2 . @PFUPA
iqutions @PFTVAD @PFUIA nd @PFUPA yieldD

|A (1) x,h | ≤ |f x,h (t x,1 ) -g n (t + x,1 )| t x,2
x-h

(t -t x,1 )|ϕ x,h (t)| dt + 1 2 t x,2
x-h

(t -t x,1 ) 2 |f x,h (θ x,1 ) -g n (θ + x,1 )||ϕ x,h (t)| dt ≤ K ∞ α 1 h + 3 2 R 1 sup 1≤i≤n d i,n K ∞ 2h sup 1≤i≤n d 2 i,n + K ∞ h α 1 + 4K ∞ R 2 K ∞ 6h sup 1≤i≤n d 3 i,n ≤ 2 3 α 1 + 3 4 R 1 + 2 3 R 2 K 2 ∞ h 2 sup 1≤i≤n d 3 i,n . @PFUQA imilrly we otinD A (2) x,h ∆ = x+h t x,N Tn (f x,h (t) -g n (t)) -(f x,h (t x,N Tn ) -g n (t x,N Tn )) ϕ x,h (t) dt |A (2) x,h | ≤ 2 3 α 1 + 3 4 R 1 + 2 3 R 2 K 2 ∞ h 2 sup 1≤i≤n d 3 i,n . @PFURA husD |A (1)
x,h + A

(2)

x,h | ≤ 4 3 α 1 + 3 2 R 1 + 4 3 R 2 K 2 ∞ h 2 sup 1≤i≤n d 3 i,n .
por i = 2, . . . , N Tn -2D similr lultions s those leding to @PFTTA giveD

t x,i+1 t x,i (f x,h (t) -g n (t)) -(f x,h (t x,i ) -g n (t x,i )) ϕ x,h (t) dt ≤ K 2 ∞ 4h 2 R 1 d 2 x,i N Tn j=1 d 2 x,j + 2K 2 ∞ 3h ( α 1 h + R 2 )d 3 x,i . husD N Tn -2 i=2 t x,i+1 t x,i (f x,h (t) -g n (t)) -(f x,h (t x,i ) -g n (t x,i )) ϕ x,h (t) dt ≤ 4 3 α 1 + R 1 + 4 3 R 2 K 2 ∞ h sup 1≤i≤n d 2 i,n . @PFUSA 2.7. PROOFS UU henD iqutions @PFUQAD @PFURA nd @PFUSA yieldD ||f x,h -P Tn f x,h || 2 ≤ 4 3 α 1 + 3 2 R 1 + 4 3 R 2 K 2 ∞ h sup 1≤i≤n d 2 i,n + 4 3 α 1 + R 1 + 4 3 R 2 K 2 ∞ h 2 sup 1≤i≤n d 3 i,n = 8 3 α 1 + 5 2 R 1 + 8 3 R 2 K 2 ∞ h sup 1≤i≤n d 2 i,n Third case. uppose now tht t x,1 = x -h nd t x,N Tn > x + h @respetively t x,1 < x -h nd t x,N Tn = x + hAF vet T n-1 = T n -{x -h} @respetively T n-1 = T n -{x + h}AF ine P T n-1 f x,h ∈ V Tn we otinD ||f x,h -P Tn f x,h || 2 ≤ ||f x,h -P T n-1 f x,h || 2 ,
we n then pply the result of the seond se to the right side of the previous inequlityF he proof of roposition PFRFQ is ompleteF Proof of Proposition 2.4.4.

he lines of this proof re sed on the work of ks nd lvisker @IWTTAF ueeping iqution @PFSRA in mind we dedue tht iqution @PFISA is equivlent toD

lim n→∞ N 2 Tn h ||f x,h -P |Tn f x,h || 2 ≥ 1 12 α(x) 1 -1 K 2/3 (t)dt 3 .
@PFUTA e shll tke the sme nottion s in the previous proofF vet g n = P |Tn f x,h D it is shown y iqution @PFIHPA in the eppendix thtX

g n (t i ) = f x,h (t i ) = n j=1 R(t j , t i )m x,h (t j ), for i = 1, • • • , n.
e hve from @pIA in the eppendix thtD

||f x,h -g n || 2 = 1 0 (f x,h (t) -g n (t))ϕ x,h (t) dt - n i=1 m x,h (t i )(f x,h (t i ) -g n (t i )) = x+h x-h (f x,h (t) -g n (t))ϕ x,h (t)dt.
uppose (rst tht t x,1 = x -h nd t x,N Tn = x + hD then the lst equlities giveD

||f x,h -g n || 2 = N Tn -1 i=1 t x,i+1 t x,i (f x,h (t) -g n (t))ϕ x,h (t)dt.
@PFUUA nder essumptions (A) nd (B)D the funtion f x,h is twie di'erentile t every t ∈ [0, 1] nd g n is twie di'erentile t every t ∈ [0, 1] exept on T n D howeverD it hs left nd right derivtivesF e expnd (f x,h -g n ) in ylor series round t x,i for t ∈]t x,i , t x,i+1 [ up to order P we otinD

f x,h (t) -g n (t) = (f x,h (t x,i ) -g n (t x,i )) + (t -t x,i )(f x,h (t x,i ) -g n (t + x,i )) + 1 2 (t -t x,i ) 2 (f x,h (σ x,t ) -g n (σ + x,t )), for some σ x,t ∈]t x,i , t[F ine g n (t x,i ) = f x,h (t x,i ) thenD f x,h (t) -g n (t) = (t -t x,i )(f x,h (t x,i ) -g n (t + x,i )) + 1 2 (t -t x,i ) 2 (f x,h (σ x,t ) -g n (σ + x,t )), @PFUVA
yn the one hndD we hve for i ∈ 1, . . . , N Tn -1D

f x,h (t x,i+1 ) -g n (t x,i+1 ) = d x,i (f x,h (t x,i ) -g n (t + x,i )) + 1 2 d 2 x,i (f x,h (σ x,i ) -g n (σ + x,i )). for some σ x,i ∈]t x,i , t x,i+1 [F husD f x,h (t x,i ) -g n (t + x,i ) = - 1 2 d x,i (f x,h (σ x,i ) -g n (σ + x,i )). @PFUWA
yn the other hndD it is shown y @pRA in the eppendix thtD

f x,h (t) -g n (t + ) = -α(t)ϕ x,h (t) + R (0,2) (•, t + ), f x,h -g n . @PFVHA
snjeting @PFUWA nd @PFVHA in @PFUVA givesD

f x,h (t) -g n (t) = - 1 2 (t -t x,i )d x,i (f x,h (σ x,i ) -g n (σ + x,i )) + 1 2 (t -t x i ) 2 (f x,h (σ x,i ) -g n (σ + x,i )) = 1 2 d x,i (t -t x,i )α(σ x,i )ϕ x,h (σ x,i ) - 1 2 (t -t x,i ) 2 α(σ x,t )ϕ x,h (σ x,t ) - 1 2 d x,i (t -t x,i ) R (0,2) (•, σ + x,i ), f x,h -g n + 1 2 (t -t x,i ) 2 R (0,2) (•, σ + x,t ), f x,h -g n . husD t x,i+1 t x,i (f x,h (t) -g n (t))ϕ x,h (t) dt = 1 2 d x,i α(σ x,i )ϕ x,h (σ x,i ) t x,i+1 t x,i (t -t x,i )ϕ x,h (t) dt - 1 2 t x,i+1 t x,i (t -t x,i ) 2 α(σ x,t )ϕ x,h (σ x,t )ϕ x,h (t) dt - 1 2 d x,i R (0,2) (•, σ + x,i ), f x,h -g n t x,i+1 t x,i (t -t x,i )ϕ x,h (t) dt + 1 2 t x,i+1 t x,i (t -t x,i ) 2 R (0,2) (•, σ + x,t ), f x,h -g n ϕ x,h (t) dt = 1 4 d 3 x,i α(σ x,i )ϕ 2 x,h (σ x,i ) - 1 6 d 3 x,i α(σ x,i )ϕ 2 x,h (σ x,i ) + 1 2 d x,i α(σ x i )ϕ x,h (σ x,i ) t x,i+1 t x,i (t -t x,i )[ϕ x,h (t) -ϕ x,h (σ x,i )] dt - 1 2 α(σ x,i )ϕ x,h (σ x,i ) t x,i+1 t x,i (t -t x,i ) 2 [ϕ x,h (t) -ϕ x,h (σ x,i )] dt - 1 2 t x,i+1 t x,i (t -t x,i ) 2 [α(σ x,t )ϕ x,h (σ x,t ) -α(σ x,i )ϕ x,h (σ x,i )]ϕ x,h (t) dt - 1 2 d x,i R (0,2) (•, σ + x,i ), f x,h -g n t x,i+1 t x,i (t -t x,i )ϕ x,h (t) dt + 1 2 t x,i+1 t x,i (t -t x,i ) 2 R (0,2) (•, σ + x,t ), f x,h -g n ϕ x,h (t) dt = 1 12 d 3 x,i α(σ x,i )ϕ 2 x,h (σ x,i ) + A (1)
x,i -A

(2)

x,i -A

x,i -A

x,i + A

(5)

x,i , @PFVIA 2.7. PROOFS UW whereD A

(1)

x,i = 1 2 d x,i α(σ x,i )ϕ x,h (σ x,i ) t x,i+1 t x,i (t -t x,i )[ϕ x,h (t) -ϕ x,h (σ x,i )] dt. A (2) x,i = 1 2 α(σ x,i )ϕ x,h (σ x,i ) t x,i+1 t x,i (t -t x,i ) 2 [ϕ x,h (t) -ϕ x,h (σ x,i )] dt. A (3) x,i = 1 2 t x,i+1 t x,i (t -t x,i ) 2 [α(σ x,t )ϕ x,h (σ x,t ) -α(σ x,i )ϕ x,h (σ x,i )]ϕ x,h (t) dt. A (4) x,i = 1 2 d x,i R (0,2) (•, σ + x,i ), f x,h -g n t x,i+1 t x,i (t -t x,i )ϕ x,h (t) dt. A (5) x,i = 1 2 t x,i+1 t x,i (t -t x,i ) 2 R (0,2) (•, σ + x,t ), f x,h -g n ϕ x,h (t) dt.
e shll now ontrol these quntitiesF vetD

B (1)
x,i = sup

t x,i <s,t<t x,i+1 |ϕ x,h (t) -ϕ x,h (s)| nd B (2)
x,i = sup

t x,i <s,t<t x,i+1 |α(t)ϕ x,h (t) -α(s)ϕ x,h (s)|. ine α nd ϕ x,h re vipshitz thenD sup 0≤i≤n B (1) x,i = O 1 h 2 sup 0≤j≤n d j,n nd sup 0≤i≤n B (2) x,i = O 1 h 2 sup 0≤j≤n d j,n . @PFVPA ilementry lultions show thtD |A (1) x,i | ≤ a 1 h B (1)
x,i d 3 x,i , |A

(2)

x,i | ≤ a 2 h B (1)
x,i d 3

x,i nd |A

(3)

x,i | ≤ a 3 h B (2)
x,i d 3 x,i , @PFVQA for pproprite onstnts a 1 , a 2 nd a 3 F e otin from the guhyEhwrtz inequlityD esE sumption (C) nd roposition PFRFQ thtD

|A (4) x,i | + |A (5) x,i | ≤ a 4 h d 3 x,i ||f x,h -g n || ≤ 1 h d 3 x,i a 4 C h a h sup 0≤j≤n d j,n , @PFVRA
for n pproprite onstnt a 4 @C is de(ned in roposition PFRFQAF husD

t x,i+1 t x,i (f x,h (t) -g n (t))ϕ x,h (t) dt = 1 12 d 3 x,i α(σ x,i )ϕ 2 x,h (σ x,i ) + A (1)
x,i -A

(2)

x,i -A

(3)

x,i -A (4) x,i + A (5) x,i ≥ 1 12 d 3 x,i α(σ x,i )ϕ 2 x,h (σ x,i ) -d 3 x,i ( a 1 h B (1) x,i + a 2 h B (2) x,i + a h h sup 0≤j≤n d j,n ). @PFVSA vetD ρ h,N Tn = sup 0≤i≤N Tn ( a 1 h B (1) x,i + a 2 h B (2) x,i + a h h sup 0≤j≤n d j,n ).
iqution @PFVPA implies tht for n pproprite onstnt c nd c we hveD

|ρ h,N Tn | ≤ c h 3 sup 0≤j≤n d j,n + c h 3/2 sup 0≤j≤n d j,n .

VH

Chapter 2 sing @PFVSA nd @PFUUA together with iqution @PFVSA in @PFUUA we otinD

||f x,h -g n || 2 ≥ N Tn -1 i=1 1 12 α(σ x,i )ϕ 2 x,h (σ x,i ) -ρ h,N Tn d 3 x,i ≥ 1 12 N Tn -1 i=1 α(σ x,i )ϕ 2 x,h (σ x,i )d 3 x,i - cN Tn h 3 sup 0≤j≤n d 4 j,n - c N Tn h 3/2 sup 0≤j≤n d 4 j,n . @PFVTA
hen the rölder9s inequlity givesD

||f x,h -g n || 2 ≥ 1 12(N Tn -1) 2 N Tn -1 j=1 [α(σ x,i )ϕ 2 x,h (σ x,i )] 1 3 d x,i 3 - cN Tn h 3 sup 0≤j≤n d 4 j,n - c N Tn h 3/2 sup 0≤j≤n d 4 j,n .
e shll now ontrol the (rst term of the right side of this inequlityF e hveD

N Tn -1 j=1 α(σ x,i )ϕ 2 x,h (σ x,i ) 1 3 d x,i 3 = x+h x-h α(t)ϕ 2 x,h (t) 1 3 dt - N Tn -1 j=1 t x,i+1 t x,i (α(t)ϕ 2 x,h (t)) 1 3 -(α(σ x,i )ϕ 2 x,h (σ x,i )) 1 3 3 = x+h x-h α(t)ϕ 2 x,h (t) 1 3 dt 3 - N Tn -1 j=1 t x,i+1 t x,i (α(t)ϕ 2 x,h (t)) 1 3 -(α(σ x,i )ϕ 2 x,h (σ x,i )) 1 3 3 -3 x+h x-h α(t)ϕ 2 x,h (t) 1 3 dt 2 N Tn -1 j=1 t x,i+1 t x,i (α(t)ϕ 2 x,h (t)) 1 3 -(α(σ x,i )ϕ 2
x,h (σ x,i ))

1 3 + 3 x+h x-h α(t)ϕ 2 x,h (t) 1 3 dt N Tn -1 j=1 t x,i+1 t x,i (α(t)ϕ 2 x,h (t)) 1 3 -(α(σ x,i )ϕ 2 x,h (σ x,i )) 1 3 2 ∆ = x+h x-h α(t)ϕ 2 x,h (t) 1 3 dt 3 + B,
e otin using @PFSIA nd the ft tht α is vipshitzD

B = O N Tn h 5/3 sup 0≤j≤n d 2 j,n 3 + O N Tn h 5/3 sup 0≤j≤n d 2 j,n h 2/3 + O N Tn h 5/3 sup 0≤j≤n d 2 j,n 2 h 1/3 .
essumption (E) implies tht for n pproprite onstnt c we hveD

|B| ≤ c N Tn h sup 0≤j≤n d 2 j,n .
2.7. PROOFS VI sing the iemnn integrility of α nd ϕ x,h we getD

||f x,h -g n || 2 ≥ 1 12(N Tn -1) 2 x+h x-h α(t)ϕ 2
x,h (t)

1 3 dt 3 - c N Tn h sup 0≤j≤n d 2 j,n - cN Tn h 3 sup 0≤j≤n d 4 j,n - c N Tn h 3/2 sup 0≤j≤n d 4 j,n ≥ 1 12N 2 Tn x+h x-h α(t)ϕ 2
x,h (t)

1 3 dt 3 - c N Tn h sup 0≤j≤n d 2 j,n - cN Tn h 3 sup 0≤j≤n d 4 j,n - c N Tn h 3/2 sup 0≤j≤n d 4 j,n = 1 12h 2 N 2 Tn x+h x-h α(t)K 2 ( x -t h ) 1 3 dt 3 - c N Tn h sup 0≤j≤n d 2 j,n - cN Tn h 3 sup 0≤j≤n d 4 j,n - c N Tn h 3/2 sup 0≤j≤n d 4 j,n = h 12N 2 Tn 1 -1 α(x -th)K 2 (t) 1 3 dt 3 - c N Tn h sup 0≤j≤n d 2 j,n - cN Tn h 3 sup 0≤j≤n d 4 j,n - c N Tn h 3/2 sup 0≤j≤n d 4 j,n .
essumption (E) implies thtD

lim n→∞ 1 h 2 N Tn sup 0≤j≤n d 2 j,n = 0, lim n→∞ 1 h 4 sup 0≤j≤n d 4 j,n N 3 Tn = 0 nd lim n→∞ c h 3/2 sup 0≤j≤n d 4 j,n N Tn = 0.
pinlly the ontinuity of α yieldsD

lim n→∞ N 2 Tn h ||f x,h -g n || 2 ≥ 1 12 α(x) 1 -1 K(t) 2 3 dt 3 .
snequlity @PFUTA is then proved for sequene of designs ontining x -h nd x + hF gonsider now ny sequene of designs {T n , n ≥ 1} stisfying essumption (E) we n djoin the points {x -h, x + h} to T n @if they ren9t presentAF rene we form sequene {S n , n ≥ 1} with S n ∈ D n+2 nd stisfying @PFUTAF e hveD

||f x,h -P |Sn f x,h || 2 ≤ ||f x,h -P |Tn f x,h || 2 . henD N 2 Sn ||f x,h -P |Sn f x,h || 2 ≤ N 2 Sn ||f x,h -P |Tn f x,h || 2 .
@PFVUA e know tht N Sn ∈ {N Tn + 1, N Tn + 2}D repling N Sn in the right term of @PFVUA y (N Tn + 2) @or @N Tn + 1A A givesD

N 2 Sn h ||f x,h -P |Sn f x,h || 2 - (4 + 2N Tn ) h ||f x,h -P |Tn f x,h || 2 ≤ N 2 Tn h ||f x,h -P |Tn f x,h || 2 .
essumption (E) nd iqution @PFSSA yieldD

lim n→∞ (4 + 2N Tn ) h ||f x,h -P |Tn f x,h || 2 = 0.
reneD for ny sequene {T n , n ≥ 1} we hveD Proof of Proposition 2.4.5.

lim n→∞ N 2 Tn h ||f x,h -P |Tn f x,h || 2 ≥ 1 12 α(x) 1 -1 K 2/3 (t)dt
yn the one hndD roposition PFRFQ yields tht there exists onstnt c > 0 suh thtD

0 ≤ σ 2 x,h m -Var ĝpro n (x) ≤ c mh sup 0≤j≤n d 2 j,n .
vemm PFRFI implies tht there exists onstnt c > 0 suh thtD

sup 0≤j≤n d 2 j,n ≤ c n 2 . husD for n ≥ 1 we hveD 0 ≤ σ 2 x,h m -Var ĝpro n (x) ≤ c c mn 2 h . pinllyD tking C = cc we otinD lim n→∞ mn 2 h σ 2 x,h m -Var ĝpro n (x) ≤ C.
snequlity @PFITA is then provedF yn the other hndD roposition PFRFR yieldsD

mN 2 Tn h σ 2 x,h m -Var ĝpro n (x) ≥ 1 12 α(x) 1 -1 K 2/3 (t)dt 3 .
vemm PFRFI implies tht there exists onstnt c > 0 suh thtD

N Tn < c nh, whih implies thtD c mn 2 h σ 2 x,h m -Var ĝpro n (x) ≥ 1 12 α(x) 1 -1 K 2/3 (t)dt 3 . pinllyD tking C = 1 12c α(x) 1 -1 K 2/3 (t)dt 3 we otinD lim n→∞ mn 2 h σ 2 x,h m -Var ĝpro n (x) ≥ C .
his onludes the proof of roposition PFRFSF Proof of Proposition 2.4.6 he (rst prt of this proof is the sme s tht of roposition @PFRFRAF ell thtD

m σ 2 x,h m -Var ĝpro n (x) = ||f x,h || 2 -||P |Tn f x,h || 2 = ||f x,h -P |Tn f x,h || 2 .
sing @PFUUA nd @PFVIA we otinD

Var ĝpro n (x) - σ 2 x,h m = - 1 m ||f x,h -P |Tn f x,h || 2 = - 1 m N Tn i=1 1 12 d 3 x,i α(σ x,i )ϕ 2 x,h (σ x,i ) + A (1)
x,i -A

(2)

x,i -A

(3)

x,i -A (4)

x,i + A

(5)

x,i , @PFVVA 2.7. PROOFS VQ for some σ

x,i ∈]t x,i , t x,i+1 [ nd some σ x,t ∈]t x,i , t[D whereD A (1) x,i = 1 2 d x,i α(σ x,i )ϕ x,h (σ x,i ) t x,i+1 t x,i (t -t x,i )[ϕ x,h (t) -ϕ x,h (σ x,i )] dt. A (2) x,i = 1 2 α(σ x,i )ϕ x,h (σ x,i ) t x,i+1 t x,i (t -t x,i ) 2 [ϕ x,h (t) -ϕ x,h (σ x,i )] dt. A (3) x,i = 1 2 t x,i+1 t x,i (t -t x,i ) 2 [α(σ x,t )ϕ x,h (σ x,t ) -α(σ x,i )ϕ x,h (σ x,i )]ϕ x,h (t) dt. A (4) x,i = 1 2 d x,i R (0,2) (•, σ + x,i ), f x,h -g n t x,i+1 t x,i (t -t x,i )ϕ x,h (t) dt. A (5) x,i = 1 2 t x,i+1 t x,i (t -t x,i ) 2 R (0,2) (•, σ + x,t ), f x,h -g n ϕ x,h (t) dt.
prom the de(nition of the regulr sequene of designs @see he(nition PFRFIA nd the men vlue theorem we hve for i = 1,

• • • , N Tn D d x,i = t x,i+1 -t x,i = F -1 i + 1 n -F -1 i n = 1 nf (t * x,i )
,

where t * x,i ∈]t x,i , t x,i+1
[F sing this together with @PFVVA we otinD

Var ĝpro n (x) - σ 2 x,h m = - 1 12mn 2 N Tn i=1 d x,i 1 f 2 (t * x,i ) α(σ x,i )ϕ 2 x,h (σ x,i ) - 1 m N Tn i=1 A (1)
x,i -A

(2)

x,i -A

(3)

x,i -A (4)

x,i + A

(5)

x,i .

vemm PFRFI yields tht N Tn = O(nh)F sing @PFVQAD @PFVRA nd @PFVPA we otinD

A (1) x,i = O 1 n 4 h 3 , A (2) x,i = O 1 n 4 h 3 , A (3) x,i = O 1 n 4 h 3 nd A (4) x,i + A (5) x,i = O 1 n 4 h 3/2 . pinllyD Var ĝpro n (x) - σ 2 x,h m = - 1 12mn 2 N Tn i=1 d x,i 1 f 2 (t * x,i ) α(σ x,i )ϕ 2 x,h (σ x,i ) + O 1 mn 3 h 2 + 1 mn 3 √ h .
sing lssil pproximtion of sum y n integrl @see for instneD vemm P in fenelE mdni et al. @PHIWA nd the ft tht 0 < h < 1 we otinD

Var ĝpro n (x) - σ 2 x,h m = - 1 12mn 2 x+h x-h α(t) f 2 (t) ϕ 2 x,h (t) dt + O 1 mn 3 h 2 .
his onludes the proof of roposition PFRFTF VR Chapter 2 Proof of Theorem 2.4.1. pirstD note tht sine α nd f re vipshitz funtions then the symptoti expression of the integrl in @PFIVA isX

1 mn 2 x+h x-h α(t) f 2 (t) ϕ 2 x,h (t)dt = 1 mn 2 h 1 -1 α(x -th) f 2 (x -th) K 2 (t) dt = 1 mn 2 h α(x) f 2 (x) 1 -1 K 2 (t) dt + 1 -1 α(x -th) f 2 (x -th) - α(x) f 2 (x) K 2 (t) dt = 1 mn 2 h α(x) f 2 (x) 1 -1 K 2 (t) dt + O 1 mn 2 .
his lst equlity together with roposition PFRFT nd roposition PFRFP onludes the proof of heorem PFRFIF Proof of Corollary 2.4.1.

vet

I 1 = 1 0 R(x, x)w(x) dx nd putD Ψ(h, m) = - C K h 2m 1 0 α(x)w(x) dx + 1 4 h 4 B 2 1 0 [g (x)] 2 w(x) dx.
e hve from heorem PFRFID

IMSE(h) = I m + Ψ(h, m) + o h 4 + h m + O 1 mn 2 h + h n + 1 n 2 h 2 ,
vet h * e s de(ned y @PFPHAF st is ler tht h * = argmin 0<h<1 Ψ(h, m) so tht Ψ(h, m) ≥ Ψ(h * , m) for every 0 < h < 1F vet h n,m e s de(ned in gorollry PFRFIF e hveD

IMSE(h * ) IMSE(h n,m ) = I 1 m + Ψ(h * , m) + o h * 4 + h * m + O 1 mn 2 h * + h * n + 1 n 2 h * 2 I 1 m + Ψ(h n,m , m) + o h 4 n,m + hn,m m + O 1 mn 2 hn,m + hn,m n + 1 n 2 h 2 n,m ≤ I 1 + mΨ(h n,m , m) + o mh * 4 + h * + O 1 n 2 h * + mh * n + m n 2 h 2 I 1 + mΨ(h n,m , m) + o mh 4 n,m + h n,m + O 1 n 2 hn,m + mhn,m n + m n 2 h 2 n,m
.

e hveD using the de(nition of h * , mh 3 n,m = O(1)D lim n,m→∞ h n,m = 0 nd using the ssumption

m n = O(1) s n, m → ∞ we know tht mΨ(h n,m , m) = O(h n,m )F husD lim n,m→∞ IMSE(h * ) IMSE(h n,m ) ≤ 1.
his onludes the proof of gorollry PFRFIF

√ m ĝpro n,m (x) -g(x) = √ m ĝpro n,m (x) -E ĝpro n,m (x) + √ m E(ĝ pro n,m (x)) -g(x)
.

(2.89) ine lim n,m→∞

√ mh = 0D n m = O(1) s n, m → ∞ nd lim n,m→∞ nh 2 = ∞ then emrk PFRFP implies thtD lim n,m→∞ √ m E(ĝ pro n,m (x)) -g(x) = 0.
gonsider now the (rst term of the right side of @PFVWAF ine

Y (t x,i ) -E(Y (t x,i )) = ε(t x,i )D we hveD s done y primn nd srirren @IWWIAD √ m ĝpro n,m (x) -E ĝpro n,m (x) = 1 √ m m j=1 n i=1 m x,h (t i )ε j (t i ) = 1 √ m m j=1 n i=1 m x,h (t i ) ε j (t i ) -ε j (x)) + n i=1 m x,h (t i ) 1 √ m m j=1 ε j (x) .
@PFWHA e strt y ontrolling the seond term of this lst equtionF sing vemm PFRFP together with vemm PFRFI we otinD

m x,h (t i,n ) =                1 2 ϕ x,h (t i,n )(t i+1,n -t i-1,n ) + O 1 n 2 h 2 + 1 n 2 √ h if i / ∈ {1, n} nd [t i-1,n , t i+1,n ] ∩ [x -h, x + h] = ∅, O 1 n 2 h 2 + 1 n 2 √ h if i ∈ {1, n}, O 1 n 2 √ h otherwise. ell tht N Tn = Card I x,h = Card {i = 1, • • • , n : [t i-1 , t i+1 ]∩]x -h, x + h[ = ∅} nd denote y t x,i the points of T n for whih i ∈ I x,h D vemm PFRFI yields tht N Tn = O(nh)F husD n i=1 m x,h (t i ) = 1 2 N Tn -1 i=2 ϕ x,h (t x,i )(t x,i+1 -t x,i-1 ) + O 1 nh .
ine lim n→∞ nh = +∞D then using the iemnn integrility of KD we otinD

lim n,m→∞ n i=1 m x,h (t i ) = 1 2 lim n,m→∞ N Tn -1 i=2 ϕ x,h (t x,i )(t x,i+1 -t x,i-1 ) = 1 -1 K(t) dt = 1.
he gentrl vimit heorem for iFiFdF vriles yieldsD

1 √ m m j=1 ε j (x) D -→ m→∞ Z where Z ∼ \(0, R(x, x)).
e shll prove now tht the (rst term of iqution @PFWHA tends to H in proility s n, m tends to in(nityF vetD

A m,n (x) = 1 √ m m j=1 n i=1 m x,h (t i ) ε j (t i ) -ε j (x) ∆ = 1 √ m m j=1
T n,j (x).

VT

Chapter 2

prom the gheyshev inequlityD it su0es to prove tht lim n,m→∞

E(A 2 m,n (x)) = 0F e hve for j = lD E(ε j (x)ε l (y)) = 0 then E(T n,j (x)T n,l (x)) = 0F reneD E(A 2 m,n (x)) = 1 m m j=1 m l=1 E(T n,j (x)T n,l (x)) = 1 m m j=1 E(T 2 n,j (x)).
e hveD

E(T 2 n,j (x)) = n i=1 n k=1 m x,h (t i )m x,h (t k )E ε j (t i ) -ε j (x) ε j (t k ) -ε j (x) = n i=1 n k=1 m x,h (t i )m x,h (t k ) R(t i , t k ) -R(t i , x) -R(x, t k ) + R(x, x) .
xote tht E(T 2 n,j (x)) does not depend on j heneD

E(A 2 m,n (x)) = n i=1 n k=1 m x,h (t i )m x,h (t k ) R(t i , t k ) -R(t i , x) -R(x, t k ) + R(x, x) ∆ = B n,1 (x) -B n,2 (x) -B n,3 (x) + B n,4 (x). @PFWIA
sing vemm PFRFP nd the pproximtion of sum y n integrl @seeD for instneD vemm P in fenelmdni et al. @PHIWA we otinD

B n,1 (x) = x+h x-h x+h x-h ϕ x,h (s)ϕ x,h (t)R(s, t) ds dt + O 1 nh = σ 2 x,h + O 1 nh .
sing iqution @PFIWA we otinD

B n,1 (x) = R(x, x) - 1 2 α(x)C K h + o(h) + O 1 nh .
where

C K = 1 -1 1 -1 |u -v|K(u)K(v)dudv. ine lim n→∞ h = 0 nd lim n→∞ nh = ∞ thenD lim n→∞ B n,1 (x) = R(x, x). @PFWPA
gonsider now the term B n,2 (x)F e otin using vemm PFRFP nd the pproximtion of sum y n integrlD

B n,2 (x) = x+h x-h x+h x-h ϕ x,h (s)ϕ x,h (t)R(s, x) ds dt + O 1 nh = x+h x-h ϕ x,h (s)R(s, x) ds + O 1 nh = 1 -1 K(s)R(x -hs, x) ds + O 1 nh = 0 -1 K(s)R(x -hs, x) ds + 1 0 K(s)R(x -hs, x) ds + O 1 nh . 2.7. PROOFS VU por s ∈] -1, 0[D ylor expnsion of R(•, x) round x yieldsD R(s, x) = R(x -sh, x) -shR (1,0) (x + , x) + o(h). imilrly for s ∈]0, 1[ we otinD R(x -sh, x) = R(x, x) -shR (1,0) (x -, x) + o(h). husD B n,2 (x) = R(x, x) -hR (1,0) (x + , x) 0 -1 s k(s) ds -hR (1,0) (x -, x) 1 0 s k(s) ds + o(h) + O 1 nh . reneD lim n→∞ B n,2 (x) = R(x, x). @PFWQA imilrlyD lim n→∞ B n,3 (x) = R(x, x). @PFWRA
st is esy to verify thtD lim n→∞ B n,4 (x) = R(x, x). @PFWSA snserting @PFWPAD @PFWQAD @PFWRA nd @PFWSA in @PFWIA yieldsD

lim n,m→∞ E(A 2 m,n (x)) = 0.
his onludes the proof of heorem PFRFQF Proof of Theorem 2.5.2. vet x ∈]0, 1[F yn the one hndD we hve from roposition PFRFT nd emrk PFRFQD

Var ĝpro n (x) = σ 2 x,h m - A 12mn 2 h α(x) f 2 (x) + O 1 mn 3 h 2 + 1 mn 2 , @PFWTA
where A = 1 -1 K 2 (t) dtF yn the other hndD it n e seen from the proof of roposition IFQFP thtD

Var ĝGM n (x) = σ 2 x,h m + O 1 mn 2 + 1 mn 3 h 2 .
@PFWUA iqutions @PFWTA nd @PFWUA then yieldD

mn 2 h Var ĝGM n -Var ĝpro n = A 12 α(x) f 2 (x) + O h + 1 nh . ell tht α(x) > 0 nd tht 1 f (x) > 0F ine h → 0 nd nh → ∞ s n, m → ∞ we otinD lim n,m→∞ mn 2 h Var ĝGM n (x) -Var ĝpro n (x) = A 12 α(x) f 2 (x) > 0.
his onludes the proof of heorem PFSFPF VV Chapter 2

Proof of Theorem 2.5.3.

e hve from the proof of roposition PFRFP @iqution @PFSQAA for ny

x ∈]0, 1[D E(ĝ pro n,m (x)) -g(x) = I h (x) -g(x) + O 1 n 2 h , @PFWVA whereD I h (x) = x+h x-h ϕ x,h (
s)g(s) ds.

reneD using @PFWTA nd @PFWVA we get for positive density mesure wD

IMSE(ĝ pro n ) = 1 m 1 0 σ 2 x,h w(x) dx - A 12mn 2 h 1 0 α(x) f 2 (x) w(x) dx + 1 0 I h (x) -g(x) 2 w(x) dx + O 1 n 4 h 2 + h n 2 + 1 mn 3 h 2 + 1 mn 2 . @PFWWA
st n e seen from roposition IFQFI thtD

E(ĝ GM n,m (x)) -g(x) = I h (x) -g(x) + O 1 n 2 h . @PFIHHA
sing @PFWUA nd @PFIHHA yieldD

IMSE(ĝ GM n ) = 1 m 1 0 σ 2 x,h w(x) dx + 1 0 I h (x) -g(x) 2 w(x) dx + O 1 n 4 h 2 + h n 2 + 1 mn 2 + 1 mn 3 h 2 .
@PFIHIA henD iqutions @PFWWA nd @PFIHIA yieldD

mn 2 h IMSE (ĝ GM n ) -IMSE (ĝ pro n ) = A 12 1 0 α(x) f 2 (x) w(x) dx + O m n 2 h + mh 2 + h + 1 nh . ine m n = O(1) nd mh 2 → 0 s n, m → ∞ we otinD lim n,m→∞ mn 2 h IMSE (ĝ GM n ) -IMSE (ĝ pro n ) = A 12 1 0 α(x) f 2 (x) w(x) dx > 0.
his onludes the proof of heorem PFSFQF

Appendix

Supplementary facts @pIA vet f e de(ned y @PFRAF e shll prove tht if

g ∈ V Tn D iFeFD if g(•) = n j=1 a j R(t j , •) for some a i ∈ RD then ||f -g|| 2 = 1 0 ϕ(s)(f (s) -g(s)) ds - n i=1 a i (f (t i ) -g(t i )). 2.8. APPENDIX VW sn ftD ||f -g|| 2 = f -g, f -g = f, f -g -g, f -g
yn the one hndD note tht f -g ∈ F(ε) nd y using @PFQA we otinD

g, f -g = n i=1 a i R(t i , •), f -g = n i=1 a i (f (t i ) -g(t i )).
yn the nother hndD vemm PFPFI nd its proof yield tht

f (•) = E(Xε(•)) where X ∈ L 2 (ε) nd thtD lim l→∞ E(X l -X) 2 = 0 where X l = l-1 j=1 (x j+1,l -x j,l )ϕ x,h (x j,l )ε(x j,l ), where (x j,l ) j=1,••• ,l is suitle prtition of [0, 1]F vet F l (•) = E(X l ε(•)) whih is n element of F(ε)F glerlyD f, f -g = f -F l , f -g + F l , f -g . e hveD | f -F l , f -g | ≤ ||f -F l || ||f -g|| ≤ E((X l -X) 2 )||f -g||. hus lim l→∞ f -F l , f -g = 0F sn dditionD F l , f -g = l-1 j=1 (x j+1,l -x j,l )ϕ(x j,l )R(x j,l , •), f -g = l-1 j=1 (x j+1,l -x j,l )ϕ(x j,l ) R(x j,l , •), f -g = l-1 j=1 (x j+1,l -x j,l )ϕ(x j,l )(f (x j,l ) -g(x j,l )). reneD lim l→∞ F l , f -g = 1 0 ϕ(t)(f (t) -g(t)) dt. pinllyD f, f -g = 1 0 ϕ(t)(f (t) -g(t)) dt.
@pPA por x ∈ [0, 1]D let f x,h e de(ned y @PFPAF e shll prove thtD

mVar(ĝ pro n (x)) = ||P |Tn f x,h || 2 .
sn ftD y the de(nition of the projetion opertor P |Tn D we hve

P |Tn f x,h ∈ V Tn nd for t ∈ [0, 1]D P |Tn f x,h (t) = n i=1 a i R(t i , t) = E( n i=1 a i (t i ) (t)) for some a i ∈ R for i = 1, • • • , n, WH Chapter 2 nd thenD ||P |Tn f x,h || 2 = E n i=1 a i (t i ) 2 = n i=1 a i n j=1 a j R(t i , t j ) = n i=1 a i P |Tn f x,h (t i ).
ell tht m x,h |Tn = f x,h |Tn R |Tn nd using @PFTA we otinD

P |Tn f x,h (t i ) = f x,h (t i ) = n j=1 m x,h (t j )R(t i , t j ).
@PFIHPA e hve thenD using @PFIHPAD

||P |Tn f x,h || 2 = n i=1 a i n j=1 m x,h (t j )R(t i , t j ) = n j=1 m x,h (t j ) n i=1 a i R(t i , t j ) = n j=1 m x,h (t j ) n i=1 m x,h (t i )R(t i , t j ) = mVar(ĝ pro n (x)).
@pQA e shll now prove prove tht every funtion in

F(ε) is ontinuous on [0, 1]F sn ft let g ∈ F(ε)D iFeFD g(•) = E(U ε(•)) for some U ∈ L 2 (ε).
por s, t ∈ [0, 1]D @PFQA nd guhyEwrtz inequlity yieldsD

|g(t) -g(s)| = | R(•, t), g -R(•, s), g | = | R(•, t) -R(•, s), g | ≤ ||R(•, t) -R(•, s)|| ||g|| = ||R(•, t) -R(•, s)|| E(U 2 ).
ine ε is of seond order proess then

E(U 2 ) < ∞ nd sine R is ontinuous on [0, 1] 2 we otinD lim s→t ||R(•, t) -R(•, s)|| 2 = lim s→t (R(t, t) + R(s, s) -2R(s, t)) = 0, whih yields tht lim s→t |g(t) -g(s)| = 0F rene g is ontinuousF @pRA uppose tht R veri(es essumptions (A), (B) nd (C)F vet f e de(ned y @PFRAF e shll prove tht if g ∈ V Tn D iFeFD g(•) = n j=1 a j R(t j , •) with (a i ) i ∈ R thenD f (t) -g (t + ) = -α(t)ϕ(t) + R (0,2) (•, t + ), f -g .
sn ftD we hveD s in iqution @PFQWAD

f (t) = -α(t)ϕ(t) + 1 0 R (0,2) (s, t + )ϕ(s) ds.
sn dditionD we hve lerly

g (t + ) = n j=1 a j R (0,2) (t j , t + ). 2.8. APPENDIX WI husD f (t) -g (t + ) = -α(t)ϕ(t) + 1 0 R (0,2) (s, t + )ϕ(s) ds - n j=1 a j R (0,2) (t j , t + ).
e hveD

R (0,2) (•, t + ), f -g = R (0,2) (•, t + ), f -R (0,2) (•, t + ), g
yn the one hndD sine y essumption @gAD R (0,2) (•, t + ) is in F(ε) then @PFQA yieldsD

R (0,2) (•, t + ), g = n j=1 a j R (0,2) (•, t + ), R(•, t j ) = n j=1
a j R (0,2) (t j , t + ). @PFIHQA yn the other hndD from vemm PFPFI we hve

f (•) = E(Xε(•)) where X ∈ L 2 (ε) ndD lim l→∞ E(X l -X) 2 = 0 with X l = l-1 j=1 (x j+1,l -x j,l )ϕ(x j,l )ε(x j,l ),
where

(x j,l ) j=1,••• ,l is suitle prtition of [0, 1]F vet F l (•) = E(X l ε(•)) ∈ F(ε)D we hveD R (0,2) (•, t + ), f = R (0,2) (•, t + ), f -F l + R (0,2) (•, t + ), F l , @PFIHRA ndD | R (0,2) (•, t + ), f -F l | ≤ ||R (0,2) (•, t + )|| ||f -F l || = ||R (0,2) (•, t + )|| E((X l -X) 2 ).
he lst ound together with essumption (C) gives lim

l→∞ | R (0,2) (•, t + ), f -F l | = 0, in dditionD R (0,2) (•, t + ), F l = l-1 j=1 (x j+1,l -x j,l )ϕ(x j,l ) R (0,2) (•, t + ), ε(x j,l ) = l-1 j=1 (x j+1,l -x j,l )ϕ(x j,l )R (0,2) (x j,l , t + ). husD lim l→∞ R (0,2) (•, t + ), F l = 1 0 ϕ(s)R (0,2) (s, t + ) ds.
@PFIHSA pinllyD using @PFIHQAD @PFIHRA nd @PFIHSA yieldD

R (0,2) (•, t + ), f -g = 1 0 ϕ(s)R (0,2) (s, t + ) ds - n j=1 a j R (0,2) (t j , t + ).
Chapter 3

Trapezoidal rule and sampling designs for the nonparametric estimation of the regression function in models with correlated errors

Abstract: he prolem of estimting the regression funtion in (xed design models with orrelted oservtions is onsideredF uh oservtions re otined from severl experimentl unitsD eh of them forms time seriesF fsed on the trpezoidl ruleD we propose simple kernel estimtor nd we derive the symptoti expression of its integrted men squred error swi nd its symptoti normlityF he prolems of the optiml ndwidth nd the optiml design with respet to the symptoti swi re lso investigtedF pinllyD simultion study is onduted to study the performne of the new estimtor nd to ompre it with the lssil estimtor of qsser nd wüller in (nite smple setF sn dditionD we study the roustness of the optiml design with respet to the misspei(tion of the utoovrine funtionF Key words: Nonparametric regression, optimal design, autocovariance function, trapezoidal rule, asymptotic normality.

Résumé: ve prolème d9estimtion de l fontion de régression est onsidéréD dns un modèle ve des erreurs orréléesF ves oservtions sont otenues à prtir de plusieurs unités expériE mentlesD hune forme une série temporelleF xous proposons un nouvel estimteur à noyuD en se snt sur l règle des trpèzesF xous étudions son omportement symptotique et nous montrons s normlité symptotiqueF ve prolème de l fenêtre optiml et l9éhntillonnge optiml sont investigués dns un ontexte symptotiqueF pinlementD nous 'etions une étude de simultion (n de tester l performne de l9estimteur proposéD pour des petites tilles d9éhntillonngeF in outreD nous étudions l roustesse de l9éhntillonnge optimle pr rpE port à l muvise spéi(tion de l fontion d9utoovrineF

Mot clés: Régression non parametrique, plans d'échantillonnage optimale, fonction d'autocovariance, règle des trapèzes, normalité asymptotique.

Introduction

e lssil prolem in ttistis is the nonprmetri estimtion of the regression funtion of response vrile Y given n explntory vrile XD iFeD estimting the funtion g de(ned y g(t) = E(Y |X = t)D sed on the oservtions of (X i , Y i ) 1≤i≤n whih re opies of (X, Y )F hese oservtions re often modeled s followsX Y i = g(t i ) + ε i where g is the unknown regression funtion to e estimtedD the {t i , i = 1, • • • , n} is the smpling design nd {ε i , i = 1, • • • , n} re entered errorsF ypilly when (ε i ) i re iFiFdF the estimtion of g hs een extensively investigted y severl uthorsF e mentionD mong othersD the work of riestly nd gho @IWUPAD fenedetti @IWUUA nd qsser nd wüller @IWUWD IWVRAF roweverD onsidering tht the oservtions re independent is not lwys relisti ssumptionF sn phrmokinetis for instneD one wishes to estimte the onentrtionEtime of some injeted mediine in the orgnismD sed on the oservtion of lood tests over period of timeF st is ler tht the oservtions provided from the sme individul re orreltedF por this resonD we shll investigte in this hpter the nonprmetri regression estimtion prolem where the oservtions re orreltedF e onsider the soElled (xed design regression model with repeted mesurementsD iFeFD

Y j (t i ) = g(t i ) + ε j (t i ) for i = 1, • • • , n nd j = 1, • • • , m, @QFIA
where {ε j , j = 1, • • • , m} is sequene of iFiFdF entered error proesses with the sme distriution s proess εF uh models re well known in growth urve nlysis nd in dose response urvesF hey n e otinedD s noted y ezzlini @IWVRAD from m individul eing oserved on period of timeF qenerllyD oservtions etween di'erent individuls will e unorreltedF reneD it is of interest to relx the ssumption of orreltion etween the experimentl unitsF wüller @IWVRA onsidered wodel @QFIA for m = 1 @oservtions on one experimentl unitA nd he supposed thtD for s = tD the ovrine Cov(ε j (t), ε j (s)) tends to 0 s n tends to in(nityD whih is not relisti ssumptionD s indited y rrt nd herly @IWVTAD in the growth urve prolemsF hey investigted the estimtion of g in wodel @QFIA with sttionry error proessF hey used the estimtor proposed y qsser nd wüller @IWUWAD nd they showed thtD in order to otin the onsisteny of the kernel estimtor in the presene of orreltionsD it is neessry to tke m experimentl units nd to let m tends to in(nityF he sttionrity ssumption is however restritiveD for instneD in the previous phrmokiE netis exmpleD it is ler tht the onentrtion of the mediine will e high t the eginning then dereses with timeF por thisD we shll investigte the estimtion of g in wodel @QFIA where ε is nonsttionry error proessF his se ws prtilly investigted y perreir et al. @IWWUA nd fenhenni nd hdi @PHHUAD where the qsser nd wüller estimtor ws usedF sn this hpterD we propose new estimtor for the regression funtion g s n pproximtion of the kernel estimtor sed on ontinuous oservtions in the whole intervl [0, 1] onstruted through stohsti integrlF eeD for instneD flnke nd fosq @PHHVAD hidi nd vouni @PHIQAF hen only disrete oservtions re villeD we use the 4est4 pproximtion of the stohsti integrlD whih is otined y using the trpezoidl rule sed on disrete oservtions t pproprite n smpling points generted y smpling density in the intervl [0, 1]F his estimtor hs reltively simpler expression thn the kernel estimtor proposed y qsser nd wüller @IWUWAF woreoverD sine this lst one depends on n integrls of kernel t middle smplesY nd my e sujet to numeril @omputtionlA instilityD for instne when qussin kernel is usedD wheres the proposed estimtor depends only on the oservtions nd the vlues of the kernel t the smpling pointsF sn ddition to its simple expressionD the proposed estimtor llows to ring n nswer to nother importnt nd open sttistil prolem under orrelted errorsD whih is the optiml 3.2. MODEL AND ESTIMATOR WS design prolemF por instneD in the previous phrmokineti exmpleD one wishes to (nd the est moments for the lood testing to e mde in order to hve etter estimte of the onentrtion urveF he optiml design prolem hs een extensively studied in prmetri regressionF e menE tion the work of ks nd lvisker @IWTTAD felouni nd fenhenni @PHISA nd more reently hette et al. @PHITA mong othersF sn the nonprmetri seD wüller @IWVRA introdued the optiml design points when the errors re symptotilly independentF re used regulr design sequene generted y density funtion f D iFeD t i = F -1 ( i n )D where F is the distriution funE tion ssoited to f F re derived the optiml design generted y density tht minimizes the symptoti sntegrted wen qured irror @swiAF o the est of our knowledgeD there exists no result onerning the prolem of optiml design for nonprmetri regression estimtion in models under more generl lss of error proessesF e lso investigte the prolem of the symptoti optiml ndwidthF e mentionD for the nonprmetri seD the work of rrt nd herly @IWVTA nd fenhenni nd hdi @PHHUAF por results on the rek down of some dt sed methods for ndwidth seletion in the presene of orreltionD for instne the ross vlidtionD nd other lterntive methodsD the reder is referred to ghiu @IWVWAD eltmn @IWWHAD rrt @IWWID IWWRA mong othersF his hpter is orgnized s followsF sn etion PD we present the new estimtor of the regression funtion g in wodel @QFIA where ε is entered error proessF sn etion QD we give the symptoti expressions of the isD the vrine nd the swiF e then derive the symptoti optiml ndwidth with respet to the symptoti swiF sn dditionD we otin the optiml design density with respet to the symptoti swiD nd we prove tht it is minimx optimlF e lso prove the symptoti normlity of the proposed estimtorF sn etion RD we ondut simultion study to investigte the performne of the new estimtor nd then to ompre it with tht of qsser nd wüllerF e lso onduted study to ompre the uniform nd the optiml smpling designsD nd to study the roustness of the optiml designD with respet to the misspei(tion of the utoovrine funtionF ine the lssil ross vlidtion riteri turned out to e ine0ient in the presene of orreltionD we use the ndwidth tht minimizes the ext swiD the omprison is performed for di'erent numers of experimentl units nd design pointsF pinllyD etion S is dedited to the proofs of our theoretil resultsF 3.2 Model and estimator e onsider m experimentl unitsD eh of them hving n di'erent mesurements of the response @sy 0

≤ t 1 < t 2 < • • • < t n ≤ 1AF he soElled (xed design regression model is de(ned s followsX Y j (t i ) = g(t i ) + ε j (t i ) where j = 1, . . . , m nd i = 1, . . . , n, @QFPA
where g is the unknown regression funtion on [0, 1] nd {ε j (t), t ∈ [0, 1]} j is sequene of error proessesF e ssume tht g ∈ C 2 ([0, 1]) nd tht (ε j ) j re iFiFdF proesses with the sme distriution s entered seond order proess εF e denote y R its utoovrine funtionF 3.2.1 Simple estimator and sampling design sn order to motivte the onstrution of our new estimtorD we onsider the regression model using m ontinuous experimentl unitsD iFeD

Y j (t) = g(t) + ε j (t) for t ∈ [0, 1] nd j = 1, • • • , m.
@QFQA e ontinuous kernel estimtor of g in wodel @QFQA is given for ny

x ∈ [0, 1] yD ĝ[0,1] (x) = 1 0 ϕ x,h (t)Y (t) dt with Y (t) = 1 m m j=1 Y j (t), @QFRA where ϕ x,h (t) = 1 h K x-t h
for kernel K nd ndwidth hF por detils on the uernel estimtion of the regression funtion sed on ontinuous oservtions seeD for instneD flnke nd fosq @PHHVA or hidi nd vouni @PHIQAF sn the prtil se where we only hve ess to disrete oservtionsD we pply the trpeE zoidl rule to pproximte the ontinuous uernel estimtor given y @QFRAF e onstrut then new simple estimtor of the regression funtion tht we shll ll the trpezoidl estimtorF fefore introduing the proposed estimtorD we egin with de(ning sequene of designs whih will e used in its onstrutionF his lss of designs ws onsidered y ks nd lvisker @IWUHAF Denition 3.2.1 Let F be a distribution function of some density f satisfying inf

t∈[0,1] f (t) > 0 and sup t∈[0,1] f (t) < ∞.
The so-called regular sequence of designs generated by a density f is dened by,

T n = t i,n = F -1 i n , i = 1, . . . , n. for n ≥ 1.
uh sequene of designs veri(es the next useful lemmF Lemma 3.2.1 For n ≥ 1 let T n = (t i,n ) i=1,••• ,n be a regular sequence of designs generated by some density function. Let x ∈]0, 1[, h > 0 and note by N Tn ∆ = Card (T n ∩[x-h, x+h]). Suppose that N Tn = 0 and that nh ≥ 1. Then,

sup 0≤j≤n (t j+1,n -t j,n ) = O 1 n
and N Tn = O(nh).

@QFSA e shll now give the de(nition of the trpezoidl estimtorD otined from disrete pE proximtion of the ontinuous estimtor ĝ[0,1] given y @QFRAF Denition 3.2.2 The trapezoidal estimator of the regression function g based on the observations

(t i,n , Y j (t i,n )) 1≤i≤n 1≤j≤m , where T n = (t i,n ) 1≤i≤n is a regular sequence of designs generated by a density function f of support intersecting [x -h, x + h] is given, for any x ∈ [0, 1], by, ĝtrap n (x) = 1 2n N Tn -1 k=1 ϕ x,h f Y (t x,k ) + ϕ x,h f Y (t x,k+1 ) , @QFTA where t x,1 < • • • < t x,N Tn are the points of T n in [x -h, x + h], ϕ x,h (t) = 1 h K x-t h , Y is given in @QFRA, K is a kernel of support [-1, 1] and h = h(n, m) is a bandwidth with 0 < h < 1.
sn order to derive our symptoti resultsD the following ssumptions on the utoovrine funtion R nd the kernel K re requiredF 3.2.2 Assumptions @eA he utoovrine funtion R exists nd is ontinuous on the squre [0, 1] 2 . @fA et the digonl @when t = s in the unit squreAD R hs ontinuous left nd right (rstEorder derivtivesD tht isX

R (0,1) (t, t -) = lim s↑t ∂R(t, s) ∂s nd R (0,1) (t, t + ) = lim s↓t ∂R(t, s) ∂s .
he jump funtion long the digonl α(t)

∆ = R (0,1) (t, t -) -R (0,1) (t, t +
) is ssumed to e ontinuous nd not identilly equl to zeroF @gA y' the digonl @when t = s in the unit squreAD R is ssumed to hve ontinuous mixed prtil derivtives up to order two ndD

A (i,j) ∆ = sup 0≤t =s≤1 |R (i,j) (t, s)| < ∞ for i, j suh tht 0 ≤ i + j ≤ 2. @hA he uernel K is even t lest in C 2 ([-1, 1]) nd K is vipshitz on EIDIF
ixmples of proesses with utoovrines stisfying essumptions (A), (B) nd (C) re given s followsF

Example 3.2.1 1. The Wiener process with autocovariance function R(s, t) = σ 2 min(s, t), has a constant jump function α(t) = σ 2 and R (i,j) (s, t) = 0 for all i, j such that i + j = 2 and s = t.

2. The Ornstein-Uhlenbeck process with a stationary autcovariance R(s, t) = σ 2 exp(-λ|s-t|) for σ > 0 and λ > 0. For this process α(t) = 2σ 2 λ and R (0,2) (s, t) = σ 2 λ 2 exp(-λ|s -t|).

3. A generalization of the Ornstein-Uhlenbeck process to a process with a nonstationary autocovariance function of the form: R(s, t) = σ 2 ρ |s λ -t λ |/λ for σ > 0, λ > 0 and 0 < ρ < 1.

For this process the jump function, which is not constant when λ = 1, is given by α(t) = -2σ 2 ln(ρ)t λ-1 . 4. Sacks and Ylvisaker (1966) gave another general class of convex stationary autcovariance functions of the form,

R(s, t) = 1/|t-s| 0 (1 -µ|t -s|)p(µ) dµ,
where p is a probability density and p its derivative are such that,

lim µ→∞ µ 3 p(µ) < ∞, and ∞ a (µp (µ) + 3p(µ)) 2 )µ 6 dµ < ∞,
for some nite constant a. For this autocovariance function, α(t) = 2 ∞ 0 µp(µ) dµ for all t.

he following kernels stisfy essumption (D)F

Example 3.2.2 1. The Quadratic kernel dened by

K(u) = 15 16 (1 -u 2 ) 2 1 {|u|≤1} . 2. The Triweight kernel dened by K(u) = 35 32 (1 -u 2 ) 3 1 {|u|≤1} .

Asymptotic results

he following propositions give the symptoti expressions of the is nd the vrine of the trpezoidl estimtor s de(ned y @QFTAF Proposition 3.3.1 Suppose that Assumption (D) is satised. Moreover assume that f ∈ C 2 ([0, 1]) and f , g are Lipschitz functions on

[0, 1]. If lim n→∞ h = 0 and lim n→∞ nh = ∞ then for any x ∈]0, 1[, Bias(ĝ trap n (x)) = 1 2 h 2 g (x)B + o(h 2 ) + O 1 n 3 h 3 , where B = 1 -1 t 2 K(t) dt.
Proposition 3.3.2 Suppose that Assumptions (A), (B), (C) and (D) are satised. Moreover assume that f ∈ C 2 ([0, 1]) and for any t ∈ [0, 1], f and R (0,2) (t, .) are all Lipschitz on [0, 1]. If

lim n→∞ h = 0 and lim n→∞ nh = ∞ then for any x ∈]0, 1[, Var(ĝ trap n (x)) = 1 m R(x, x) - h 2 C K α(x) + V 12mn 2 h α(x) f 2 (x) + o h m + O 1 mn 2 + 1 mn 3 h 3 , where V = 1 -1 K 2 (t) dt and C K = 1 -1 1 -1 |u -v|K(u)K(v)dudv.
ropositions QFQFI nd QFQFP llow to derive the symptoti expression of the men squred error @wiA of the rpezoidl estimtor @QFTAF he integrted men squred error @swiA is then otined y integrting the wi with respet to some weight funtion wF he results re nnounedD without proofD in the following theoremF Theorem 3.3.1 If all the assumptions of Propositions 3.3.1 and 3.3.2 are satised then for any

x ∈]0, 1[, MSE(ĝ trap n (x)) = 1 m R(x, x) - h 2 α(x)C K + V 12mn 2 h α(x) f 2 (x) + 1 4 h 4 [g (x)] 2 B 2 + o h 4 + h m + O 1 n 3 h + 1 mn 3 h 3 + 1 mn 2 + 1 n 6 h 6 , IMSE(ĝ trap n ) = 1 m 1 0 R(x, x) - h 2 α(x)C K w(x) dx + V 12mn 2 h 1 0 α(x) f 2 (x) w(x) dx + 1 4 h 4 B 2 1 0 [g (x)] 2 w(x) dx + o h 4 + h m + O 1 n 3 h + 1 mn 3 h 3 + 1 mn 2 + 1 n 6 h 6 , (3.7)
where w is a continuous density function, V , B and C K are given in Propositions 3.3.1, 3.3.2. he previous heorem showsD the e0ieny of the rpezoidl estimtorD sine the swi tends to 0 when m → ∞D h → 0 nd nh → ∞ s n → ∞F he symptoti optiml ndwidth is otined y minimizing the symptoti swi s given y the following propositionF Proposition 3.3.3 (Optimal bandwidth) Suppose that the assumptions of Theorem 3.3.1 are satised. Moreover assume that m n = O(1) as n, m → ∞. Denote by IMSE(h) the IMSE of the trapezoidal estimator when the bandwidth h is used. Then the bandwidth,

h * = C K 1 0 α(x)w(x) dx 2B 2 1 0 [g (x)] 2 w(x) dx 1/3 m -1/3 , @QFVA
is optimal in the sense that,

lim n,m→∞ IMSE(h * ) IMSE(h n,m ) ≤ 1,
for any sequence of bandwidths h n,m verifying:

lim n,m→∞
h n,m = 0 and lim n,m→∞

mh 3 n,m < +∞,
where B and C K are given in Propositions 3.3.1 and 3.3.2.

e re interested now in (nding the optiml design densityD iFeD funtion f * ording to the riteri f * ∈ argmin f swiD where the minimum is tken with respet to the lss of positive densities de(ned on [0, 1]F sn view of heorem QFQFID the symptoti optiml design density veri(esD

f * ∈ argmin f >0, 1 0 f (x)dx=1 1 0 α(x) f 2 (x) w(x) dx.
his optimiztion prolem is solved in the following orollryF Corollary 3.3.1 (Optimal design) Suppose that the assumptions of Theorem 3.3.1 are satised. If lim n→∞ nh 2 = ∞ and lim n,m→∞ n m = ∞, then the optimal sampling density with respect to the asymptotic IMSE is given by,

f * (t) = {α(t)w(t)} 1/3 1 0 {α(s)w(s)} 1/3 ds 1 [0,1] (t).
@QFWA vet ĝtrap n,f * e the rpezoidl estimtor @QFTA with f = f * de(ned y @QFWAF e hveD

IMSE(ĝ trap n,f * ) = 1 m 1 0 R(x, x) - 1 2 α(x)C K h w(x) dx + V 12mn 2 h 1 0 (α(x)w(x)) 1/3 dx 3 + 1 4 h 4 B 2 1 0 [g (x)] 2 w(x) dx + o h 4 + h m + O 1 n 3 h + 1 mn 3 h 3 + 1 mn 2 + 1 n 6 h 6 .
Remark 3.3.1 Let ĝtrap n,unif be the Trapezoidal estimator @QFTA with a uniform density, i.e, f = f unif the identity in [0, 1]. The asymptotic IMSE of ĝtrap n,unif is given by,

IMSE(ĝ trap n,unif ) = 1 m 1 0 R(x, x) - 1 2 α(x)C K h w(x) dx + V 12mn 2 h 1 0 α(x)w(x) dx + 1 4 h 4 B 2 1 0 [g (x)] 2 w(x) dx + o h 4 + h m + O 1 n 3 h + 1 mn 3 h 3 + 1 mn 2 + 1 n 6 h 6 .

IHH Chapter 3

The reduction of the residual IMSE, IMSE ∆ = IMSE -σ 2

x,h /m, by using the asymptotic optimal design over the uniform design is then,

rIM SE = IMSE(ĝ trap n,unif ) -IMSE(ĝ trap n,f * ) IMSE(ĝ trap n,unif ) ∼ 1 - 1 0 (α(x)w(x)) 1/3 dx 3 1 0 α(x)w(x) dx .
For instance, if R(s, t) = st min(s, t) then α(t) = t 2 . Taking w ≡ 1 gives rIM SE ∼ 35%.

pinllyD the next theorem gives the symptoti normlity of the rpezoidl estimtor @QFTAF he optiml designD generted y the density funtion @QFWAD my not e roust with respet to the missspei(tion of the utoovrine jump funtion αD nd the weight funtion wF por thisD we shll use minimx riterion to otin the optiml smpling designF fiedermnn nd hette @PHHIA gve the following riterionD density funtion f * is sid to e minimx optiml ifD

f * ∈ argmin f >0, 1 0 f (t)dt=1 max (α,w)∈Λ Ψ (α,w) (f ), @QFIHA whereD Ψ (α,w) (f ) = 1 0 α(t) f 2 (t) w(t)dt, ndD Λ = (α, w) ∈ (C[0, 1]) 2 1 0 α(t)dt < 1 , 1 0 w(s) 1/2 ds 2 < 2 .
he following theorem ssures tht the symptoti optiml design densityD de(ned in gorollry QFQFID is optiml in the sense of minimxF Theorem 3.3.2 (Minimax optimality) Suppose that the assumptions of Theorem 3.3.1 are satised. The function f * given by @QFWA is optimal with respect to the minimax criterion @QFIHA.

pinllyD we onlude our theoretil results y the symptoti normlity of the trpezoidl estiE mtorD presented in the following theoremF Theorem 3.3.3 (Asymptotic normality) Suppose that the assumptions of Theorem 3.3.1 are satised. If

lim m→∞ √ mh 2 = 0 and lim n→∞ nh 2 = ∞ then for any x ∈]0, 1[, √ m ĝtrap n (x) -g(x) D -→ Z, with Z ∼ N (0, R(x, x)),
where D denotes the convergence in distribution and N is the normal distribution.

3.4 Simulation study 3.4.1 Performance of the estimator sn this setionD we investigte the performne of our estimtor @QFTA in (nite smple setF e shll use the ui growth urveD used y rrt nd herly @IWVTA nd fenhenni nd hdi @PHHUAD g(x) = 10x 3 -15x 4 + 6x 5 for 0 < x < 1. @QFIIA IHI his funtion ws minly used due to its similrity to the logisti funtion whih is frequently found in growth urve nlysisF he smpling points re tken to eX

t i = (i -0.5)/n for i = 1, • • • , n. @QFIPA
he error proess ε is tken to e the iener error proess with utoovrine funtion R(s, t) = σ 2 min(s, t)F he uernel used here is the qudrti kernel given y K(u) = (15/16)(1u 2 ) 2 I [-1,1] (u)F he ndwidth used in this study is the optiml ndwidth with respet to the ext IMSEF e onsider the men of ll estimtors otined from IHH simultionsF e tke σ 2 = 0.5 nd simultions for other vlues of σ 2 gve similr resultsD they re given in pigure QFI for (xed numer of oservtions n = 100 nd three di'erent vlues of experimentl units m = 5, 20, 100F pigure QFIX gui regression funtion is in plin line nd the trpezoidl estimtor is in dshed oneF st is ler thtD the performne of the trpezoidl estimtor gets etter s m inresesF yur im now is to ompre the trpezoidl estimtor to tht of qsser nd wüller @IWUWA @referred y qw estimtorAD given for ny x ∈]0, 1[ yD

ĝGM n (x) = n i=1 m i m i-1 ϕ x,h (t)dt Y (t i ), @QFIQA
where

m 0 = 0D m n = 1 nd m i = (t i + t i+1 )/2 for i = 2, • • • , n -1D ϕ x,h (t) = (1/h)K((x -t)/h) nd Y (t i ) = (1/m) m j=1 Y j (t i
)F his omprison is onduted with respet to the nonEsymptoti swi nd under di'erent types of orreltion errorsF e onsider gin the ui regression funtionD the design given y @QFIPA nd the qudrti kernelF he two error proesses onsidered here re the sttionry yrnsteinEhlenek proess with R(s, t) = exp(-λ|s -t|)D nd the nonsttionry iener proess with R(s, t) = σ 2 min(s, t)F e investigte vrious 4mount4 of orreltion y tking di'erent vlues of oth σ 2 nd λF e tke the weight density w to e uniform on [0, 1]D nd we ompre the optiml nonE symptoti IMSE of the two estimtorsD iFeFD inf 0<h<1 IMSE(h)F he ndwidth h is hosen over grid from 0.09 to 0.5F he results re given in les QFIEQFT for n = 30 nd for di'erent vlues of mF he tles present the integrted is squred denoted y Ibias 2 D integrted vrine denoted y Ivar nd the IMSE together with the optiml ndwidth ssoited to the smllest nonEsymptoti swi for eh estimtorF he tles re orgnized ording to the 4degree4 of orreltion of the errorsF st n e seen tht the optiml ndwidth is the sme for oth estimtorsD in dditionD s expetedD it dereses s m inresesF gonsider (rst the se of strong orrelted errorsD iFeD for lrge σ 2 nd smll λF sn le QFID for the iener proess with σ 2 = 1D it ppers tht the qEw estimtor hs slightly smller Ibias 2 while the trpezoidl estimtor hs slightly smller Ivar nd sine the Ibias 2 is too smll ompred to the Ivar then the trpezoidl estimtor hs slightly smller swiF por the yrnsteinEhlenek with λ = 1 @FfF le QFPA it n e seen tht the trpezoidl estimtor hs slightly etter performne euse of smller swiD due to smller Ibias 2 nd smller IvarF gonsider now the se of moderte orrelted errorsF sn le QFQ @for the iener proess with σ 2 = 0.5A it seems tht the qEw estimtor hs slightly smller Ibias 2 while the trpezoidl estimtor hs slightly smller Ivar nd smller swiF hile for the yrnsteinEhlenek proess with λ = 25D presented in tle QFRD the qEw estimtor hs slightly smller swi due to smller Ibias 2 nd smller IvarF pinllyD onsider the wekly orrelted errorsD iFeD for smll vlue of σ 2 nd lrge vlue of λF sn tle QFSD for the iener proess with σ 2 = 0.06F it ppers tht the qEw estimtor hs slightly smller Ibias 2 while the trpezoidl estimtor hs smller Ivar nd smller swiF roweverD for the yrnsteinEhlenek proess with λ = 50 @FfF le QFTA the trpezoidl estimtor hs slightly smller Ibias 2 while the qEw estimtor hs slightly smller Ivar nd swiF yverllD the two estimtorsD iFeD the trpezoidl estimtor nd the qsser nd wüller estimE torD hve 4pproximtely4 the sme performneF reneD the proposed estimtorD whih hs simpler expressionD is s e0ient s the lssil qsser nd wüller estimtorF sn ll the previous sesD it ppers tht Ibias 2 is lwys smller thn IvarF st should e noted here thtD oth of the estimtors hve oundry prolemsF e modi(ed kernel t the edgesD s suggested y rrt nd herly @IWVTAD ws used in this simultionF 3.4.2 Optimal design enother importnt spet we looked t in this simultion study ws the use of the symptoti optiml design in (nite smple setF e onsider the lss of utoovrine funtions introdued in ixmple QFPFI s followsX

R(s, t) = σ 2 ρ |s λ -t λ |/λ , σ 2 > 0, λ > 0 nd 0 < ρ < 1,
for whih the jump funtion α(t) = -2σ 2 ln(ρ)t λ-1 F sn prtiulr when λ = 1 we otin n yrnsteinEhlenek sttionry error proessF sn our numeril studies we will onsider the nonsttionry seD λ = 1F his lss of nonsttionry prmetri utoovrine funtions ws introdued y xúñezEentón nd oodworth @IWWRA to study the e0y of ohler implntsF st ws lso used y severl other uthorsD y perreir et al. @IWWUA who were interested in otining the optiml ndwidth for the qsser nd wüller estimtorD y iemmermn et al. @IWWVAD nd then y xúñezEentón @IWWUA to study the speeh reognition dtF IHQ e ompreD for m ∈ {5, 10, 20, 30} nd for instne h = 0.123D the nonEsymptoti swi @tking w ≡ 1A of the trpezoidl estimtor @QFTAD using oth the uniform design @QFIPAD iFeFD f ≡ 1 nd the optiml design generted y f * given in @QFWAD iFeFD

f * λ (t) = λ + 2 3 t (λ-1)/3 1 [0,1] (t) nd t * λ,i = i n 3/(λ+2)
.

Robustness of the optimal design he optiml design depends on the utoovrine prmeter λD whih is not known in prtieD therefore we nnot use this design to ompute the estimte of the regression funtion gF es n lterntiveD we n estimte (rst the utoovrine prmeter λD from the oservtions otined following uniform designD then we otin the estimted optiml design de(ned s followsX

f * λ (t) = λ + 2 3 t ( λ-1)/3 1 [0,1] (t) nd t * λ,i = i n 3/( λ+2)
.

he estimtion of the utoovrine prmeters is otined y minimizing the following riteE rionD s done for instne in perreir et al.

@IWWUAX Q n,m (σ 2 , λ, ρ) = 1 n 2 n i=1 n j=1 R(t i , t j ) -R(t i , t j ) 2 , @QFIRA
where the empiril orreltion estimtor is given s followsX

R(t i , t j ) = 1 m -1 m k=1 Y k (t i ) -Y (t i ) Y k (t j ) -Y (t j ) for i, j = 1, • • • , n.
prom ememiy @IWVSAD s noted y perreir et al. @IWWUAD it is known tht the non liner lest squre estimtor ( σ 2 , λ, ρ) is onsistentF sn our simultion studyD we (xed λ = 4, σ 2 = 0.5 nd ρ = 0.5F o estimte (λ, σ 2 , ρ)D we generted 100 mtries (Y j (t i )) 1≤i≤n 1≤j≤m of oservtions using the uniform designF por every mtrixD we used the qenerlized imulted enneling @qeA lgorithm to minimize the funtion @QFIRAD the estimtion ( λ, σ 2 , ρ) is then the medin of the IHH estimted vluesF por more detils on the use of the softwre lgorithm funtionD see ing et alF @PHIQAF his lgorithm is essentilly known for its ility to hndle very omplex nonEliner ojetive funtions with very lrge numer of optimF he results re given in les QFUEQFIHD where the redution in the swi y tking the optiml design insted of the uniform design is given yD

rIM SE λ = IMSE(ĝ trap n,unif ) -IMSE(ĝ trap n,f * λ ) IMSE(ĝ trap n,unif )
, nd the redution in the swi y tking the plugEin estimted optiml design insted of the uniform design is given yD

rIM SE λ = IMSE(ĝ trap n,unif ) -IMSE(ĝ trap n,f * λ ) IMSE(ĝ trap n,unif )
.

st n e seen in les QFUEQFIH tht there exists redution of the swi of the rpezoidl estimtor when using the optiml designD even for smll vlues of the smpling size n nd the numer of experimentl units mF vikewiseD the estimted optiml design otined y estimting the ovrine prmeterD still provides redution of the swi over the uniform designF his redution is lose to the one using the theoretil optiml designD this shows tht the optiml design is roust when the ovrine prmeter hs to e estimtedF le QFIX he integrted squred isD sntegrted vrineD swi nd the optiml ndwidth in terms of m under the iener error proess with σ 2 = 1, for the qw nd the trpezoidl estimtorF

n = 20 m Ibias 2 Ivar IMSE h opt GM S PFVVQP×10 -3 VFRWTU×10 -2 VFUVSH×10 -2 HFRII T rap PFVVQQ×10 -3 VFRWSW×10 -2 VFUVRQ×10 -2 HFRII GM IS IFHRVIT×10 -3 PFWPWQ×10 -2 QFHQRI×10 -2 HFQPP T rap IFHRVST×10 -3 PFWPUT×10 -2 QFHQPS×10 -2 HFQPP GM QH PFUTWI×10 -4 IFSITW×10 -2 IFSRRT×10 -2 HFPQQ T rap PFVSQS×10 -4 IFSIPR×10 -2 IFSRHW×10 -2 HFPQQ
le QFPX he integrted squred isD sntegrted vrineD swi nd the optiml ndwidth in terms of m under the yrnsteinEhlenek error proess with λ = 1 for the qw nd the trpezoidl estimtorF

n = 20 m Ibias 2 Ivar IMSE h opt GM S RFSUHHP×10 -3 IFUHSUH×10 -1 IFUSIRH×10 -1 HFRT T rap RFSUHHI×10 -3 IFUHSTS×10 -1 IFUSIQS×10 -1 HFRT GM IS IFQIHSH×10 -3 SFVVVR×10 -2 TFHIWR×10 -2 HFQR T rap IFQHWWU×10 -3 SFVVSU×10 -2 TFHITU×10 -2 HFQR GM QH UFUVVW×10 -4 PFWVIV×10 -2 QFHSWU×10 -2 HFQH T rap UFUVPV×10 -4 PFWUWI×10 -2 QFHSTW×10 -2 HFQH
IHS le QFQX he integrted squred isD sntegrted vrineD swi nd the optiml ndwidth in terms of m under the iener error proess with σ 2 = 0.5 for the qw nd the trpezoidl estimtorF

n = 20 m Ibias 2 Ivar IMSE h opt GM S IFHRVI×10 -3 RFQWQW×10 -2 RFRWVV×10 -2 HFQPP T rap IFHRVS×10 -3 RFQWIS×10 -2 RFRWTQ×10 -2 HFQPP GM IS PFUTWI×10 -4 IFSITW×10 -2 IFSRRT×10 -2 HFPQQ T rap PFVSQS×10 -4 IFSIPR×10 -2 IFSRHW×10 -2 HFPQQ GM QH IFIUWP×10 -4 UFUPPV×10 -3 UFVRHU×10 -3 HFIVV T rap IFRIUS×10 -4 UFTUQQ×10 -3 UFVISH×10 -3
HFIVV le QFRX he integrted squred isD sntegrted vrineD swi nd the optiml ndwidth in terms of m under the yrnsteinEhlenek error proess with λ = 25 for the qw nd the trpezoidl estimtorF

n = 20 m Ibias 2 Ivar IMSE h opt GM S RFQWQI×10 -3 PFUITQ×10 -2 QFISST×10 -2 HFRSS T rap RFQWQH×10 -3 PFUITS×10 -2 QFISSV×10 -2 HFRSS GM IS IFUWRP×10 -3 IFPVIW×10 -2 IFRTIQ×10 -2 HFQTT T rap IFUWQS×10 -3 IFPVPR×10 -2 IFRTIV×10 -2 HFQTT GM QH IFHRVI×10 -3 UFHVHV×10 -3 VFIPWH×10 -3 HFQPP T rap IFHRVS×10 -3 UFHVSS×10 -3 VFIQRI×10 -3 HFQPP
le QFSX he integrted squred isD sntegrted vrineD swi nd the optiml ndwidth in terms of m under the iener error proess with σ 2 = 0.06 for the qw nd the trpezoidl estimtorF

n = 20 m Ibias 2 Ivar IMSE h opt GM S WFWUIR×10 -5 SFSUVI×10 -3 SFTUUV×10 -3 HFIVI T rap IFPVRI×10 -4 SFSQUQ×10 -3 SFTTSU×10 -3 HFIVI GM IS WFWUIR×10 -5 RFTRVR×10 -3 RFURVI×10 -3 HFIVI T rap IFPVRI×10 -4 RFTIRS×10 -3 RFURPW×10 -3 HFIVI GM QH WFWUIR×10 -4 QFWVRR×10 -3 RFHVRI×10 -3 HFIVI
T rap IFPVRI×10 -4 QFWSSP×10 -3 RFHVQT×10 -3 HFIVI le QFTX he integrted squred isD sntegrted vrineD swi nd the optiml ndwidth in terms of m under the yrnsteinEhlenek error proess with λ = 50 for the qw nd the trpezoidl estimtorF por the ske of lrityD we omit the n in t i,n F por i = 1, • • • , n -1 the wen lue heorem @mFvFtA yields tht there exists

n = 20 m Ibias 2 Ivar IMSE h opt GM S RFQRWT×10 -3 IFWWHS×10 -2 PFRPSS×10 -2 HFRSR T rap RFQRWR×10 -3 IFWWHU×10 -2 PFRPSU×10 -2 HFRSR GM IS PFVIWR×10 -3 IFVHRW×10 -2 PFHVTV×10 -2 HFRHV T rap PFVIWP×10 -3 IFVHSQ×10 -2 PFHVUP×10 -2 HFRHV GM QH PFVIWR×10 -3 IFSRUH×10 -2 IFVPWH×10 -2 HFRHV T rap PFVIWP×10 -3 IFSRUR×10 -2 IFVPWQ×10 -
η i ∈]t i , t i+1 [ suh thtD t i+1 -t i = F -1 ( i + 1 n ) -F -1 ( i n ) = 1 nf (η i ) . ine inf 0≤t≤1 f (t) > 0 then t i+1 -t i = O( 1 n
)F e shll now prove the seond prt of the vemmF ine T n ∩ [x -h, x + h] = ∅D there exist i 1 , i N indexes in {1, . . . , n} suh thtD

N Tn ≤ i N -i 1 + 1.
prom the de(nition of the regulr sequene we hve for ll i = 1, ..., nD

t i = F -1 i n thus i = nF (t i ).
sing this nd the mFvFt we otin for some

x ∈]t i 1 , t i N [D N Tn ≤ n F (t i N ) -F (t i 1 ) + 1 = n(t i N -t i 1 )f ( x ) + 1, he oundedness of f nd the ft tht t i N -t i 1 ≤ 2h yieldD N Tn ≤ (2 sup 0≤t≤1 f (t)) nh + 1.
his onludes the proof of the seond prt of vemm QFPFI sine 1 ≤ nh.

Proof of Proposition 3.3.1.

por h smll enough nd sine

T n ∩ [x -h, x + h] = ∅ we tke t x,1 < t x,2 < • • • < t x,N Tn the points of T n in [x -h, x + h]F ine E(Y (t i )) = g(t i ) for ll i = 1, • • • , n we hveD E(ĝ trap n (x)) = 1 2n N Tn -1 k=1 ϕ x,h f g (t x,k ) + ϕ x,h f g (t x,k+1
) .

3.5. PROOFS IHW prom the de(nition of the regulr sequene of designs we hve for k = 1, . . . , N Tn -1D

F (t x,k+1 ) -F (t x,k ) = 1 n ⇐⇒ t x,k+1 t x,k f (t) dt = 1 n . @QFISA husD E(ĝ trap n (x)) = 1 2 N Tn -1 k=1 t x,k+1 t x,k ϕ x,h f g (t x,k ) + ϕ x,h f g (t x,k+1 ) f (t) dt.
vetD

I h (x) = x+h x-h ϕ x,h (t)g(t) dt = N Tn -1 k=1 t x,k+1 t x,k ϕ x,h (t)g(t) dt + t x,1 x-h ϕ x,h (t)g(t) dt + x+h t x,N Tn ϕ x,h (t)g(t) dt,
nd writeD

E(ĝ trap n (x)) = E(ĝ trap n (x)) -I h (x) + I h (x) ∆ = ∆ x,h + I h (x). @QFITA e (rst ontrol ∆ x,h F vetD ∆ x,h = ∆ 1 x,h + ∆ 2 x,h , @QFIUA whereD ∆ 1 x,h = 1 2 N Tn -1 k=1 t x,k+1 t x,k ϕ x,h f g (t x,k )f (t) -ϕ x,h (t)g(t) dt - 1 2 t x,1 x-h ϕ x,h (t)g(t) dt - 1 2 x+h t x,N Tn ϕ x,h (t)g(t) dt. ∆ 2 x,h = 1 2 N Tn -1 k=1 t x,k+1 t x,k ϕ x,h f g (t x,k+1 )f (t) -ϕ x,h (t)g(t) dt - 1 2 t x,1 x-h ϕ x,h (t)g(t) dt - 1 2 x+h t x,N Tn ϕ x,h (t)g(t) dt. por t ∈ [x -h, t x,1 ]D ylor expnsion of ϕ x,h round (x -h) yieldsD ϕ x,h (t) = ϕ x,h (x -h) + (t -(x -h))ϕ x,h (x -h) + 1 2 (t -(x -h)) 2 ϕ x,h (θ x,h ), @QFIVA
for some θ x,h ∈]x -h, t x,1 [F ell tht y de(nition of ϕ x,h we hveD

sup 0≤t≤1 |ϕ (j)
x,h (t)| ≤ c j h j+1 for j = 0, 1, 2, @QFIWA for some pproprite onstnts c j where j = 0, 1, 2F sn dditionD sine

ϕ x,h is in C 2 nd of support [x -h, x + h] thenD ϕ x,h (x -h) = ϕ x,h (x + h) = ϕ x,h (x -h) = ϕ x,h (x + h) = 0. @QFPHA
sing @QFPHA nd @QFIWA in @QFIVA nd using vemm @QFPFIA we otin for t ∈

[x -h, t x,1 ]D ϕ x,h (t) = 1 2 (t -(x -h)) 2 ϕ x,h (θ x,h ) = O 1 n 2 h 3 , @QFPIA vikewiseD for t ∈ [t x,N Tn , x + h] we hveD ϕ x,h (t) = 1 2 (t -(x + h)) 2 ϕ x,h (θ x,h ) = O 1 n 2 h 3 , @QFPPA where θ x,h ∈]t x,N Tn , x + h[F reneD t x,1 x-h ϕ x,h (t)g(t) dt = O 1 n 3 h 3 nd x+h t x,N Tn ϕ x,h (t)g(t) dt = O 1 n 3 h 3 . husD ∆ 1 x,h = 1 2 N Tn -1 k=1 t x,k+1 t x,k ϕ x,h f g (t x,k ) - ϕ x,h f g (t) f (t) dt + O 1 n 3 h 3 , ndD ∆ 2 x,h = 1 2 N Tn -1 k=1 t x,k+1 t x,k ϕ x,h f g (t x,k+1 ) - ϕ x,h f g (t) f (t) dt + O 1 n 3 h 3 . ell tht ϕ x,h is in C 2 nd f, g ∈ C 2 ([0, 1])D then for ny t ∈]t x,k , t x,k+1 [ ylor expnsions of ϕ x,h f g nd f round t x,k giveD ∆ 1 x,h = 1 2 N Tn -1 k=1 ϕ x,h f g (t x,k )f (t x,k ) t x,k+1 t x,k (t x,k -t) dt - 1 2 N Tn -1 k=1 ϕ x,h f g (t x,k )f (t x,k ) t x,k+1 t x,k (t -t x,k ) 2 dt - 1 4 N Tn -1 k=1 ϕ x,h f g (t x,k ) t x,k+1 t x,k (t -t x,k ) 3 f (η x,k ) dt - 1 4 N Tn -1 k=1 f (t x,k ) t x,k+1 t x,k (t -t x,k ) 2 ϕ x,h f g (θ x,k ) dt - 1 4 N Tn -1 k=1 f (t x,k ) t x,k+1 t x,k (t -t x,k ) 3 ϕ x,h f g (θ x,k ) dt - 1 8 N Tn -1 k=1 t x,k+1 t x,k (t -t x,k ) 4 ϕ x,h f g (θ x,k )f (η x,k ) dt + O 1 n 3 h 3 , 3.5. PROOFS

III

where θ x,k nd η x,k re in ]t x,k , t[F ell tht the funtions g (j) , f (j) for j = 0, 1, 2 re ll oundedD then using@QFIWA nd vemm QFPFI we getD

N Tn -1 k=1 ϕ x,h f g (t x,k ) t x,k+1 t x,k (t -t x,k ) 3 f (η x,k ) dt = O 1 n 3 h . @QFPQA N Tn -1 k=1 f (t x,k ) t x,k+1 t x,k (t -t x,k ) 3 ϕ x,h f g (θ x,k ) dt = O 1 n 3 h 2 . @QFPRA N Tn -1 k=1 t x,k+1 t x,k (t -t x,k ) 4 ϕ x,h f g (θ x,k )f (η x,k ) dt = O 1 n 4 h 2 .
@QFPSA xote thtD sine ϕ x,h D g nd f re ll lipshitz thenD

ϕ x,h f g (θ x,k ) = ϕ x,h f g (t x,k ) + ϕ x,h f g (θ x,k ) - ϕ x,h f g (t x,k ) = ϕ x,h f g (t x,k ) + O 1 nh 4 . @QFPTA
snjeting @QFPQAD @QFPRAD @QFPSA nd @QFPTA in ∆ 1

x,h we hveD

∆ 1 x,h = 1 2 N Tn -1 k=1 ϕ x,h f g (t x,k )f (t x,k ) t x,k+1 t x,k (t x,k -t) dt - 1 2 N Tn -1 k=1 ϕ x,h f g (t x,k )f (t x,k ) t x,k+1 t x,k (t -t x,k ) 2 dt - 1 4 N Tn -1 k=1 ϕ x,h f g (t x,k )f (t x,k ) t x,k+1 t x,k (t -t x,k ) 2 dt + O 1 n 3 h 3 . vet d x,k = t x,k+1 -t x,k F e otin y si integrtionD ∆ 1 x,h = - 1 4 N Tn -1 k=1 ϕ x,h f g (t x,k )f (t x,k )d 2 x,k - 1 6 N Tn -1 k=1 ϕ x,h f g (t x,k )f (t x,k )d 3 x,k - 1 12 N Tn -1 k=1 ϕ x,h f g (t x,k )f (t x,k )d 3 x,k + O 1 n 3 h 3 .
@QFPUA imilrly we verify thtD

∆ 2 x,h = 1 4 N Tn -1 k=1 ϕ x,h f g (t x,k+1 )f (t x,k+1 )d 2 x,k - 1 6 N Tn -1 k=1 ϕ x,h f g (t x,k+1 )f (t x,k+1 )d 3 x,k - 1 12 N Tn -1 k=1 ϕ x,h f g (t x,k+1 )f (t x,k+1 )d 3 x,k + O 1 n 3 h 3 , @QFPVA IIP Chapter 3
umming @QFPUA nd @QFPVA givesD

∆ x,h = ∆ 1 x,h + ∆ 2 x,h = 1 4 N Tn -1 k=1 d 2 x,k ϕ x,h f g (t x,k+1 )f (t x,k+1 ) - ϕ x,h f g (t x,k )f (t x,k ) - 1 6 N Tn -1 k=1 d 3 x,k ϕ x,h f g (t x,k+1 )f (t x,k+1 ) + ϕ x,h f g (t x,k )f (t x,k ) - 1 12 N Tn -1 k=1 d 3 x,k ϕ x,h f g (t x,k+1 )f (t x,k+1 ) + ϕ x,h f g (t x,k )f (t x,k ) + O 1 n 3 h 3 . ine ϕ x,h is in C 1 nd g , f ∈ C 1 ([0, 1])D ylor expnsion of ϕ x,h f g f round t x,k yieldsD ϕ x,h f g f (t x,k+1 ) = ϕ x,h f g f (t x,k ) + d x,k ϕ x,h f g f (ν x,k ),
where ν x,k ∈]t x,k , t x,k+1 [F e then hveD

∆ x,h = 1 4 N Tn -1 k=1 d 3 x,k ϕ x,h f g f (ν x,k ) - 1 6 N Tn -1 k=1 d 3 x,k ϕ x,h f g (t x,k+1 )f (t x,k+1 ) + ϕ x,h f g (t x,k )f (t x,k ) - 1 12 N Tn -1 k=1 d 3 x,k ϕ x,h f g (t x,k+1 )f (t x,k+1 ) + ϕ x,h f g (t x,k )f (t x,k ) + O 1 n 3 h 3 .
prom the de(nition of the regulr sequene of designs nd using the mFvFtF we otin for k

= 1, • • • , N Tn -1, t x,k+1 t x,k f (t) dt = 1 n ⇐⇒ d x,k = 1 nf (t * x,k )
for some t * x,k ∈]t x,k , t x,k+1 [. @QFPWA his eqution yieldsD x,h , f (j) nd g (j) for j = 0, 1, 2 nd pplying vemm QFSFI in the eppendix with u

∆ x,h = 1 4n 2 N Tn -1 k=1 d x,k 1 f 2 (t * x,k ) ϕ x,h f g f (ν x,k ) - 1 6n 2 N Tn -1 k=1 d x,k ϕ x,h f g (t x,k+1 ) f (t x,k+1 ) f 2 (t * x,k ) + ϕ x,h f g (t x,k ) f (t x,k ) f 2 (t * x,k ) - 1 12n 2 N Tn -1 k=1 d x,k ϕ x,h f g (t x,k+1 ) f (t x,k+1 ) f 2 (t * x,k ) + ϕ x,h f g (t x,k ) f (t x,k ) f 2 (t * x,k ) + O 1 n 3 h 3 .
(t) = 1 f 2 (t) nd v(t) = ϕ x,h f g f (t) we otinD N Tn -1 k=1 d x,k 1 f 2 (t * x,k ) ϕ x,h f g f (ν x,k ) = x+h x-h 1 f 2 (t) ϕ x,h f g f (t) dt + O 1 nh 3 . imilrlyD tking u(t) = ϕ x,h f g (t) nd v(t) = f (t) f 2 (t) in vemm QFSFI we otinD N Tn -1 k=1 d x,k ϕ x,h f g (t x,k+1 ) f (t x,k+1 ) f 2 (t * x,k ) = x+h x-h ϕ x,h f g (t) f (t) f 2 (t) dt + O 1 nh 3 . egin tking u(t) = ϕ x,h f g (t) nd v(t) = f (t) f 2 (t) we otinD N Tn -1 k=1 d x,k ϕ x,h f g (t x,k+1 ) f (t x,k+1 ) f 2 (t * x,k ) = x+h x-h ϕ f g (t) 1 f (t) dt + O 1 nh 3 . reneD ∆ x,h = 1 4n 2 x+h x-h 1 f 2 (t) ϕ x,h f g f (t) dt - 1 3n 2 x+h x-h ϕ x,h f g (t) f (t) f 2 (t) dt - 1 6n 2 x+h x-h ϕ x,h f g (t) 1 f (t) dt + O 1 n 3 h 3 . imple derivtions yieldD ∆ x,h = 1 4n 2 x+h x-h ϕ x,h f g (t) 1 f (t) dt + 1 4n 2 x+h x-h ϕ x,h f g (t) f (t) f 2 (t) dt - 1 3n 2 x+h x-h ϕ x,h f g (t) f (t) f 2 (t) dt - 1 6n 2 x+h x-h ϕ x,h f g (t) 1 f (t) dt + O 1 n 3 h 3 = 1 12n 2 x+h x-h ϕ x,h f g (t) 1 f (t) dt - 1 12n 2 x+h x-h ϕ x,h f g (t) f (t) f 2 (t) dt + O 1 n 3 h 3 = 1 12n 2 x+h x-h ϕ x,h f g 1 f (t) dt + O 1 n 3 h 3 . pinllyD ∆ x,h = 1 12n 2 ϕ x,h f g 1 f (x + h) - ϕ x,h f g 1 f (x -h) + O 1 n 3 h 3 .
he lst eqution together with @QFPHA yieldD

∆ x,h = O 1 n 3 h 3 .
@QFQHA he ontrol of I h (x) is lssil nd it n e seen from qsser nd wüller thtD

I h (x) = g(x) + 1 2 h 2 g (x) 1 -1 t 2 K(t) dt + o(h 2 ). @QFQIA IIR Chapter 3
pinllyD olleting @QFITAD @QFQHA nd @QFQIA givesD

Bias(ĝ trap n (x)) = 1 2 h 2 g (x)B + o(h 2 ) + O 1 n 3 h 3 ,
where B = 1 -1 t 2 K(t) dtF his onludes the proof of roposition QFQFIF Proof of Proposition 3.3.2.

he gretest lines of this proof re sed on the work of felouni nd fenhenni @PHISAF por h smll enough nd sine

T n ∩ [x -h, x + h] = ∅ we hveD 0 ≤ t 1 < • • • < x -h ≤ t x,1 < • • • < t x,N Tn ≤ x + h < • • • < t n ≤ 1. vetD Φ(t, s) = ϕ x,h f (t)R(t, s) ϕ x,h f (s), ndD σ 2 x,h = x+h x-h x+h x-h ϕ x,h (t)R(t, s)ϕ x,h (s) ds dt.
@QFQPA yn the one hndD

Var(ĝ trap n (x)) = 1 4mn 2 N Tn -1 i=1 N Tn -1 j=1 Φ(t x,i , t x,j ) + Φ(t x,i , t x,j+1 ) + Φ(t x,i+1 , t x,j ) + Φ(t x,i+1 , t x,j+1 )
sing @QFISA one n writeD

Var(ĝ trap n (x)) = 1 4m N Tn -1 i=1 N Tn -1 j=1 t x,i+1 t x,i t x,j+1
t x,j Φ(t x,i , t x,j ) + Φ(t x,i , t x,j+1 ) + Φ(t x,i+1 , t x,j ) + x,i+1 , t x,j+1 ) f (s) f (t) ds dt.

yn the other hnd we hveD

σ 2 x,h = N Tn -1 i=1 N Tn -1 j=1 tx,i+1
d x,i = O( 1 n )F sing @QFPIA nd @QFPPA we hveD sup (x-h)≤t≤t x,1 |ϕ x,h (t)| = O 1 n 2 h 3 nd sup t x,N Tn ≤t≤(x+h) |ϕ x,h (t)| = O 1 n 2 h 3 .
@QFQQA ine f nd R re oundedD using @QFIWA nd @QFQQA we otinD

t x,1 x-h x+h t x,N Tn Φ(t, s)f (t) f (s) ds dt = O 1 n 6 h 6 , x+h t x,N Tn x+h t x,N Tn Φ(t, s)f (t) f (s) ds dt = O 1 n 6 h 6 , t x,1 x-h t x,1 x-h Φ(t, s)f (t) f (s) ds dt = O 1 n 6 h 6 , N Tn -1 j=1 t x,1 x-h t x,j+1 t x,j Φ(t, s)f (t) f (s) ds dt = O 1 n 3 h 3 , N Tn -1 j=1 x+h t x,N Tn t x,j+1 t x,j Φ(t, s)f (t) f (s) ds dt = O 1 n 3 h 3 . husD σ 2 x,h = N Tn -1 i=1 N Tn -1 j=1 t x,i+1 t x,i t x,j+1 t x,j Φ(t, s)f (t) f (s) ds dt + O 1 n 3 h 3 .
e shll ontrol the residul vrine Var(ĝ trap n (x)) -σ 2

x,h m F por thisD letD N i,j (t, s) = Φ(t x,i , t x,j ) + Φ(t x,i+1 , t x,j ) + Φ(t x,i , t x,j+1 ) + Φ(t x,i+1 , t x,j+1 ) -4Φ(t, s), @QFQRA nd putD

I i,j = 1 4m t x,i+1 t x,i t x,j+1 t x,j
N i,j (t, s)f (t) f (s) ds dt. @QFQSA he residul vrine n then e written s followsD

Var(ĝ trap n (x)) - σ 2 x,h m = N Tn -1 i=1 I i,i + N Tn -1 i =j=1 I i,j + O 1 mn 3 h 3 , @QFQTA
trting with the digonl terms I i,i F ine for ny s, t ∈ [0, 1]D we hve N i,i (s, t) = N i,i (t, s)D then we n writeD

I i,i = 1 2m t x,i+1 t x,i t t x,i N i,i (t, s)f (t) f (s) ds dt. @QFQUA
feuse of essumption (B)D N i,i hs left nd right (rst order derivtives on the digonl on [0, 1] 2 F por ny s, t suh tht @t x,i < s ≤ t < t x,i+1 AD ylor expnsion of Φ round (t x,i , t x,i )

givesD

Φ(t, s) = Φ(t, t x,i ) + (s -t x,i )Φ (0,1) (t, t x,i ) + 1 2 (s -t x,i ) 2 Φ (0,2) (t, η (1)
s,i ) = Φ(t x,i , t x,i ) + (t -t x,i )Φ (1,0) ( (1)
t,i , t x,i ) + (s -t x,i )Φ (0,1) ( i , t x,i )

+ (s -t x,i )(t -i )Φ (1,1) ( (2) t,i , t x,i ) + 1 2 (s -t x,i ) 2 Φ (0,2) (t, η (1)
s,i ),

for some i ∈]t x,i , t x,i+1 [D some (1) t,i in ]t x,i , t[D some (2)
t,i etween t nd i nd some η

(1)

t,i in ]t x,i , s[F e hveD Φ(t, s) = Φ(t x,i , t x,i ) + (t -t x,i )Φ (1,0) ( i , t x,i ) + (s -t x,i )Φ (0,1) ( i , t x,i ) + (t -t x,i ) Φ (1,0) ( (1) t,i , t x,i ) -Φ (1,0) ( i , t x,i ) + (s -t x,i )(t -i )Φ (1,1) ( (2) t,i , t x,i ) + 1 2 (s -t x,i ) 2 Φ (0,2) (t, η (1)
s,i ).

por l nd l integers suh tht l + l ≤ 2D essumption (C) yieldsD

sup s =t |Φ (l,l ) (t, s)| = O 1 h l+l +2 . @QFQVA sn dditionD sine ϕ x,h , ϕ x,h , 1 f , R nd R(•, t x,i ) re ll ontinuous on ]t x,i , t x,i+1 [D then fo s = t in ]t x,i , t x,i+1 [ we hveD Φ (1,0) (s, t x,i ) -Φ (1,0) (t, t x,i ) = ϕ x,h f (t x,i ) R(s, t x,i ) ϕ x,h f (s) - ϕ x,h f (t) + R (1,0) (s, t x,i ) ϕ x,h f (s) - ϕ x,h f (t) + ϕ x,h f (t) R (1,0) (s, t x,i ) -R (1,0) (t, t x,i ) + ϕ x,h f (t) R(s, t x,i ) -R(t, t x,i ) = O 1 nh 4 .
pinllyD using this eqution together with vemm QFPFI we otinD

Φ(t, s) = Φ(t x,i , t x,i ) + (t -t x,i )Φ (1,0) ( i , t x,i ) + (s -t x,i )Φ (0,1) ( i , t x,i ) + O 1 n 2 h 4 .
@QFQWA imilrly we verify thtD

Φ(t x,i+1 , t x,i+1 ) = Φ(t x,i , t x,i ) + d x,i Φ (1,0) ( i , t x,i ) + d x,i Φ (0,1) ( i , t x,i ) + O 1 n 2 h 4 , @QFRHA nd thtD Φ(t x,i+1 , t x,i ) = Φ(t x,i , t x,i ) + d x,i Φ (1,0) ( i , t x,i ) + O 1 n 2 h 4 .
@QFRIA snserting @QFQWAD @QFRHA nd @QFRIA in @QFQRA for i = j nd using @QFQVA nd vemm QFPFID we otinD

N i,i (t, s) = 3d x,i Φ (1,0) ( i , t x,i ) -4(t -t x,i )Φ (1,0) ( i , t x,i ) + d x,i Φ (0,1) ( i , t x,i ) -4(s -t x,i )Φ (0,1) ( i , t x,i ) + O 1 n 2 h 4 .
3.5. PROOFS IIU epling this expression in @QFQUAD nd using the oundedness of f nd vemm QFPFID we otinD

I i,i = 1 2m d x,i 3Φ (1,0) ( i , t x,i ) + Φ (0,1) ( i , t x,i ) t x,i+1 t x,i t t x,i f (t) f (s) ds dt -4Φ (1,0) ( i , t x,i ) t x,i+1 t x,i t t x,i (t -t x,i )f (t) f (s) ds dt -4Φ (0,1) ( i , t x,i ) t x,i+1 t x,i t t x,i (s -t x,i )f (t) f (s) ds dt + O 1 mn 4 h 4 . @QFRPA ell tht f is in C 2 ([0, 1]) nd tht d x,i = O( 1 n ) from vemm QFPFIF st n esily e veri(ed tht for ny integers l nd l X t x,i+1 t x,i t t x,i (s -t x,i ) l (s -t x,i ) l f (t) f (s) ds dt = f 2 (t x,i ) d (l+l +2) x,i (l + 1)(l + l + 2) + O 1 n l+l +3 .
sing this lst iqution together with @QFQVA in @QFRPA oveD nd @QFPWA we otinD

I i,i = 1 12m Φ (1,0) ( i , t x,i ) -Φ (0,1) ( i , t x,i ) f 2 (t x,i )d 3 x,i + O 1 mn 4 h 4 = 1 12mn 2 Φ (1,0) ( i , t x,i ) -Φ (0,1) ( i , t x,i ) f 2 (t x,i ) f 2 (t * x,i ) d x,i + O 1 mn 4 h 4 .
pinlly using vemm QFPFID the integrility of ϕ x,h , ϕ x,h , f, f nd R (0,1) (., t) nd pplying vemm QFSFI in the eppendixD we otinD

N Tn -1 i=1 I i,i = 1 12mn 2 N Tn -1 i=1 Φ (1,0) ( i , t x,i ) -Φ (0,1) ( i , t x,i ) f 2 (t x,i ) f 2 (t * x,i ) d x,i + O 1 mn 3 h 3 = 1 12mn 2 x+h x-h Φ (1,0) (t + , t) -Φ (0,1) (t + , t) dt + O 1 mn 3 h 3 . (3.43) ine Φ (0,1) (t + , t) = Φ (0,1) (t, t -) = Φ (1,0) (t -, t)D thenD N Tn -1 i=1 I i,i = - 1 12mn 2 x+h x-h Φ (1,0) (t -, t) -Φ (1,0) (t + , t) dt + O 1 mn 3 h 3 . @QFRRA
xowD it remins to hndle the o' digonl termF essumption (B) yields tht N i,j for i = j is twie di'erentile o' the digonl on [0, 1] 2 F ylor expnsion of N i,j round (t x,i , t x,j ) for i = j up to order P givesD

Φ(t, s) = Φ(t x,i , t x,j ) + (t -t x,i )Φ (1,0) (t x,i , t x,j ) + (s -t x,j )Φ (0,1) (t x,i , t x,j ) + 1 2 (t -t x,i ) 2 Φ (2,0) ( (1)
x,i , t x,j ) + 1 2 (s -t x,j ) 2 Φ (0,2) (t x,i , η

x,j )

+ (t -t x,i )(s -t x,j )Φ (1,1) ( (1)
x,i , η

x,j ), @QFRSA for some

(1)

x,i etween t x,i nd t nd some η

(1)

x,j etween t x,j nd sF king t = t x,i+1 nd s = t x,j in @QFRSAD we otinD

Φ(t x,i+1 , t x,j ) = Φ(t x,i , t x,j ) + d x,i Φ (1,0) (t x,i , t x,j ) + 1 2 d 2 x,i Φ (2,0) ( (2)
x,i , t x,j ), @QFRTA for some

(2)

x,i in ]t x,i , t x,i+1 [F king t = t x,i nd s = t x,j+1 in @QFRSAD we otinD Φ(t x,i , t x,j+1 ) = Φ(t x,i , t x,j ) + d x,j Φ (0,1) (t x,i , t x,j ) + 1 2 d 2 x,j Φ (0,2) (t x,i , η

x,j ), @QFRUA for some η

(2)

x,j in ]t x,j , t x,j+1 [F king t = t x,i+1 nd s = t x,j+1 in @QFRSAD we otinD

Φ(t x,i+1 , t x,j+1 ) = Φ(t x,i , t x,j ) + d x,i Φ (1,0) (t x,i , t x,j ) + d x,j Φ (0,1) (t x,i , t x,j ) + 1 2 d 2 x,i Φ (2,0) ( (3)
x,i , t x,j ) +

1 2 d 2 x,j Φ (0,2) (t x,i , η (3)
x,j )

+ d x,i d x,j Φ (1,1) ( (3)
x,i , η

x,j ), @QFRVA e otin y inserting @QFRSAD @QFRTAD @QFRUA nd @QFRVA in @QFQRAD N i,j (t, s) = Φ (1,0) (t x,i , t x,j ) 2d x,i -4(t -t x,i ) + Φ (0,1) (t x,i , t x,j ) 2d x,j -4(s -t x,j )

+ 1 2 d 2 x,i Φ (2,0) ( (2)
x,i , t x,j ) + Φ (2,0) (

x,i , t x,j ) -2(t -t x,i ) 2 Φ (2,0) (

x,i , t x,j )

+ 1 2 d 2 x,j Φ (0,2) (t x,i , η (2)
x,j ) + Φ (0,2) (t x,i , η

x,j ) -2(s -t x,j ) 2 Φ (0,2) (t x,i , η

x,j )

+ d x,i d x,j Φ (1,1) ( (3)
x,i , η

x,j ) -4(t -t x,i )(s -t x,j )Φ (1,1) (

x,i , η

x,j ).

e otin inserting the lst eqution in @QFQSAD I i,j = 1 4m (s -t x,j )f (t)f (s)dtds .

I

(3)

i,j = 1 2 d 2 x,i t x,i+1 t x,i t x,j+1 t x,j Φ (2,0) ( (2)
x,i , t x,j ) + Φ (2,0) (

x,i , t x,j ) f (t)f (s)dtds

-2 t x,i+1 t x,i t x,j+1 t x,j (t -t x,i ) 2 Φ (2,0) ( (1)
x,i , t x,j )f (t)f (s)dtds.

I (4) i,j = 1 2 d 2 x,j t x,i+1 t x,i t x,j+1 t x,j Φ (0,2) (t x,i , η (2)
x,j ) + Φ (0,2) (t x,i , η

x,j ) f (t)f (s)dtds

-2 t x,i+1 t x,i t x,j+1 t x,j (s -t x,j ) 2 Φ (0,2) (t x,i , η (1)
x,j )f (t)f (s)dtds.

I (5) i,j = d x,i d x,j t x,i+1 t x,i t x,j+1 t x,j Φ (1,1) ( (3)
x,i , η

x,j )f (t)f (s)dtds

-4 t x,i+1 t x,i t x,j+1 t x,j (t -t x,i )(s -t x,j )Φ (1,1) ( (1)
x,i , η

x,j )f (t)f (s)dtds.

e (rst onsider the term I

(1) i,j F por l = 0, 1, 2D letD

ω i,l = t x,i+1 t x,i (t -t x,i ) l f (t)dt @QFSHA he term I (1)
i,j n then e written sD

I (1) i,j = Φ (1,0) (t x,i , t x,j ) 2d x,i ω i,0 ω j,0 -4ω i,1 ω j,0 . @QFSIA ixpnding f round t x,i yieldsD ω i,l = t x,i+1 t x,i (t -t x,i ) l f (t x,i ) + (t -t x,i )f (t x,i ) + 1 2 (t -t x,i ) 2 f ( (4) x,i ) dt = d (l+1) x,i (l + 1) f (t x,i ) + d (l+2) x,i (l + 2) f (t x,i ) + O 1 n (l+3) , @QFSPA IPH Chapter 3
for some (4)

x,i in ]t x,i , t x,i+1 [F hus for l = 0, 1, 2D

I (1) i,j = Φ (1,0) (t x,i , t x,j ) 2d x,i d x,i f (t x,i ) + d 2 x,i 2 f (t x,i ) + O 1 n 3 × d x,j f (t x,j ) + d 2 x,j 2 f (t x,i ) + O 1 n 3 -4 d 2 x,i 2 f (t x,i ) + d 3 x,i 3 f (t x,i ) + O 1 n 4 d x,j f (t x,j ) + d 2 x,j 2 f (t x,i ) + O 1 n 3 = Φ (1,0) (t x,i , t x,j ) - 1 3 f (t x,i )f (t x,j )d 3 x,i d x,j + O 1 n 5 .
e otin using iqutions @QFQVA nd @QFPWAD I

(1)

i,j = -

1 3 Φ (1,0) (t x,i , t x,j )f (t x,i )f (t x,j )d 3 x,i d x,j + O 1 n 5 h 3 = - 1 3n 2 Φ (1,0) (t x,i , t x,j ) f (t x,i ) f 2 (t * x,i ) f (t x,j )d x,i d x,j + O 1 n 5 h 3 ,
for some t *

x,i in ]t x,i , t x,i+1 [F sing vemm QFPFI nd the integrility of ϕ x,h , ϕ x,h , f, nd of R (0,1) (., t) nd pplying vemm QFSFI twieD we otinD

N Tn -1 i =j=1 I (1) i,j = - 1 3n 2 N Tn -1 i =j=1 Φ (1,0) (t x,i , t x,j ) f (t x,i ) f 2 (t * x,i ) f (t x,j )d x,i d x,j + O 1 n 3 h = - 1 3n 2 x+h x-h x+h x-h Φ (1,0) (t, s) f (t) f 2 (t) f (s)1 {s =t} dt ds + O 1 n 3 h 2 .
@QFSQA imilrly we verify thtD

N Tn -1 i =j=1 I (2) i,j = - 1 3n 2 x+h x-h x+h x-h Φ (0,1) (t, s) f (s) f 2 (s) f (t)1 {s =t} dt ds + O 1 n 3 h 2 = - 1 3n 2 x+h x-h x+h x-h Φ (1,0) (t, s) f (t) f 2 (t) f (s)1 {s =t} dt ds + O 1 n 3 h 2 .
@QFSRA e now ontrol the term I 3 i,j F e hveD

I (3) i,j = d 2 x,i Φ (2,0) (t x,i , t x,j ) t x,i+1 t x,i t x,j+1 t x,j f (t)f (s)dtds -2Φ (2,0) (t x,i , t x,j ) t x,i+1 t x,i t x,j+1 t x,j (t -t x,i ) 2 f (t)f (s)dtds + 1 2 d 2 x,i t x,i+1 t x,i t x,j+1 t x,j Φ (2,0) ( (2)
x,i , t x,j ) + Φ (2,0) (

x,i , t x,j ) -2Φ (2,0) (t x,i , t x,j )f (t)f (s)dtds

-2 t x,i+1 t x,i t x,j+1 t x,j (t -t x,i ) 2 Φ (2,0) ( (1)
x,i , t x,j ) -Φ (2,0) (t x,i , t x,j ) f (t)f (s)dtds.

3.5. PROOFS IPI sing @QFQVAD vemm QFPFI nd iqution @QFSHA we getD

I (3) i,j = d 2 x,i Φ (2,0) (t x,i , t x,j )ω i,0 ω j,0 -2Φ (2,0) (t x,i , t x,j )ω i,2 ω i,0 + O 1 n 5 h 5 .
xote (rst thtD using @QFSPA for l = 0 long with l = 2 nd vemm QFPFID we otinD

I (3) i,j = 1 3 Φ (2,0) (t x,i , t x,j )d 3 x,i d x,j f (t x,i )f (t x,j ) + O 1 n 5 h 5 = 1 3n 2 Φ (2,0) (t x,i , t x,j ) f (t x,i ) f 2 (t * x,i ) f (t x,j )d x,i d x,j + O 1 n 5 h 5 , vikewiseD using vemm QFPFI nd the integrility of ϕ (k)
x,h , f (k) for k = 0, 1, 2 we hveD

N Tn -1 i =j=1 I (3) i,j = 1 3n 2 x+h x-h x+h x-h Φ (2,0) (t, s) f (s) f (t) 1 {s =t} dt ds + O 1 n 3 h 3 .
@QFSSA imilrlyD we otinD

N Tn -1 i =j=1 I (4) i,j = 1 3n 2 x+h x-h x+h x-h Φ (0,2) (t, s) f (t) f (s) 1 {s =t} dt ds + O 1 n 3 h 3 = 1 3n 2 x+h x-h x+h x-h Φ (2,0) (t, s) f (s) f (t) 1 {s =t} dt ds + O 1 n 3 h 3 .
@QFSTA pinllyD for the term I

(5)

i,j D we hveD

I (5) i,j = d x,i d x,j Φ (1,1) (t x,i , t x,j ) t x,i+1 t x,i t x,j+1 t x,j f (t)f (s)dtds -4Φ (1,1) (t x,i , t x,j ) t x,i+1 t x,i t x,j+1 t x,j (t -t x,i )(s -t x,j )f (t)f (s)dtds + d x,i d x,j t x,i+1 t x,i t x,j+1 t x,j Φ (1,1) ( (3)
x,i , η

x,j ) -Φ (1,1) (t x,i , t x,j ) f (t)f (s)dtds

-4 t x,i+1 t x,i t x,j+1 t x,j (t -t x,i )(s -t x,j ) Φ (1,1) ( (1)
x,i , η

x

,j ) -Φ (1,1) (t x,i , t x,j ) f (t)f (s)dtds = d x,i d x,j Φ (1,1) (t x,i , t x,j )ω i,0 ω j,0 -4Φ (1,1) (t x,i , t x,j )ω i,1 ω j,1 + d x,i d x,j t x,i+1 t x,i t x,j+1 t x,j Φ (1,1) ( (3)
x,i , η

x,j ) -Φ (1,1) (t x,i , t x,j ) f (t)f (s)dtds

-4 t x,i+1 t x,i t x,j+1 t x,j (t -t x,i )(s -t x,j ) Φ (1,1) ( (1)
x,i , η

x,j ) -Φ (1,1) (t x,i , t x,j ) f (t)f (s)dtds.

ell tht f, f , 1 f re ll ounded nd using @QFQVA nd @QFSPA with l = l = 1 we otinD

I (5) i,j = O 1 n 5 h 5 . IPP Chapter 3 pinllyD sine N Tn = O(nh) from vemm QFPFID we otinD N Tn -1 i =j=1 I (5) i,j = O 1 n 3 h 3 .
@QFSUA epling @QFSQAD @QFSRAD @QFSSAD @QFSTA nd @QFSUA in @QFRWA we otinD

N Tn -1 i =j=1 I i,j = 1 6mn 2 x+h x-h x+h x-h Φ (2,0) (t, s)f (t) -Φ (1,0) (t, s)f (t) f 2 (t) 1 {s =t} f (s)dtds + O 1 mn 3 h 3 = 1 6mn 2 x+h x-h s x-h ∂ ∂s Φ (1,0) (t, s) f (t) dt f (s)ds + O 1 mn 3 h 3 + 1 6mn 2 x+h x-h x+h s ∂ ∂s Φ (1,0) (t, s) f (t) dt f (s)ds + O 1 mn 3 h 3 = 1 6mn 2 x+h x-h Φ (1,0) (s -, s) -Φ (1,0) (s + , s) ds + O 1 mn 3 h 3 + 1 6mn 2 x+h x-h Φ (1,0) (x + h, s) f (x + h) - Φ (1,0) (x -h, s) f (x -h) f (s) ds + O 1 mn 3 h 3 . xote tht for t = sD Φ (1,0) (t, s) = ϕ x,h (t)f (t) -ϕ x,h (t)f (t) f 2 (t) R(t, s) + ϕ x,h (t) f (t) R (1,0) (t, s) ϕ x,h (s) f (s) . @QFSVA st follows from @QFPHA thtD Φ (1,0) (x + h, s) f (x + h) = Φ (1,0) (x -h, s) f (x -h) = 0 for ll s ∈]x -h, x + h[. husD N Tn -1 i =j=1 I i,j = 1 6mn 2 x+h x-h Φ (1,0) (t -, t) -Φ (1,0) (t + , t) dt + O 1 mn 3 h 3 .
@QFSWA snserting @QFRRA nd @QFSWA in @QFQTAD we otinD

Var(ĝ trap n (x)) = 1 m σ 2 x,h + 1 12mn 2 x+h x-h Φ (1,0) (t -, t) -Φ (1,0) (t + , t) dt + O 1 mn 3 h 3 . (3.60) epplying @QFSVA it follows thtD Φ (1,0) (t -, t) -Φ (1,0) (t + , t) = ϕ 2 x,h (t) f 2 (t) R (1,0) (t -, t) -R (1,0) (t -, t) = ϕ 2 x,h (t) f 2 (t) α(t). @QFTIA 3.5. PROOFS IPQ epling @QFTIA in @QFTHA we otinD Var(ĝ trap n (x)) = 1 m σ 2 x,h + 1 12mn 2 x+h x-h ϕ 2 x,h (t) f 2 (t) α(t) dt + O 1 mn 3 h 3 . @QFTPA ine α nd f re ontinuous on [0, 1]D then one n writeD x+h x-h α(t) f 2 (t) ϕ 2 x,h (t)dt = 1 h 1 -1 α(x -th) f 2 (x -th) K 2 (t) dt = 1 h α(x) f 2 (x) 1 -1 K 2 (t) dt + 1 h 1 -1 α(x -th) f 2 (x -th) - α(x) f 2 (x) K 2 (t) dt = 1 h α(x) f 2 (x) 1 -1 K 2 (t) dt + O(1). @QFTQA
ell tht for n even kernelD we hve simpli(ed expression of σ 2 x,h given y fenhenni nd hdi @PHHUA s followsD σ 2

x,h = R(x, x) -

1 2 α(x)C K h + o(h), @QFTRA where C K = 1 -1 1 -1 |u -v|K(u)K(v)dudv. pinllyD using @QFTQA nd @QFTRA in @QFTPA yieldsD Var(ĝ trap n (x)) = 1 m R(x, x) - 1 2 α(x)C K h + 1 12mn 2 h α(x) f 2 (x) 1 -1 K 2 (t) dt + o h m + O 1 mn 2 + 1 mn 3 h 3 . his onludes the proof of roposition QFQFPF Proof of Proposition 3.3.3. vet I 1 = 1 0 R(x, x)w(x) dxD I 2 = 1 0 α(x) f 2 (x) w(x) dx nd putD Ψ(h, m) = - C K h 2m 1 0 α(x)w(x) dx + 1 4 h 4 B 2 1 0 [g (x)] 2 w(x) dx.
e hve from iqution @QFUA in heorem QFQFID

IMSE(h) = I 1 m + Ψ(h, m) + V I 2 12mn 2 h + o h 4 + h m + O 1 n 3 h + 1 mn 3 h 3 + 1 mn 2 + 1 n 6 h 6 . vet h * e s de(ned in @QFVAF st is ler tht h * = argmin 0<h<1 Ψ(h, m) so tht Ψ(h, m) ≥ Ψ(h * , m)
for every 0 < h < 1F vet h n,m e s de(ned in gorollry QFQFQF e hveD

IMSE(h * ) IMSE(h n,m ) = I1 m + Ψ(h * , m) + V I2 12mn 2 h * + o h * 4 + h * m + O 1 n 3 h * + 1 mn 3 h * 3 + 1 mn 2 + 1 n 6 h * 6 I1 m + Ψ(h n,m , m) + V I2 12mn 2 hn,m + o h 4 n,m + hn,m m + O 1 n 3 hn,m + 1 mn 3 h 3 n,m + 1 mn 2 + 1 n 6 h 6 n,m ≤ I 1 + mΨ(h n,m , m) + V I2 12n 2 h * + o mh * 4 + h * + O m n 3 h * + 1 n 3 h * 3 + 1 n 2 + m n 6 h * 6 I 1 + mΨ(h n,m , m) + V I2 12n 2 hn,m + o mh 4 n,m + h n,m + O m n 3 hn,m + 1 n 3 h 3 n,m + 1 n 2 + m n 6 h 6 n,m . sing the de(nition of h * , mh 3 n,m = O(1)D lim n,m→∞ h n,m = 0 nd the ssumption m n = O(1) s n, m → ∞ we know tht mΨ(h n,m , m) = O(h n,m )F henD lim n,m→∞ IMSE(h * ) IMSE(h n,m ) ≤ 1.
his onludes the proof of roposition QFQFQF Proof of Corollary 3.3.1.

vet f * e s de(ned in @QFWAF vet

D(f ) = 1 0 α(x) f 2 (x) w(x) dx, then it is su0ient to prove thtX D(f * ) ≤ D(f ) for every positive density f on [0, 1].
epplying rölder9s inequlityD we getD

D(f * ) = 1 0 {α(x)w(x)} 1/3 dx 3 = 1 0 α(x)w(x) f 2 (x) 1/3 f 2/3 (x) dx 3 ≤ 1 0 α(x)w(x) f 2 (x) dx 1 0 f (x) dx 2 = D(f ). reneD rgmin {f >0 density on [0,1]} D(f ) = f * .
his ompletes the proof of gorollry QFQFIF Proof of Theorem 3.3.2.

vet f * e s de(ned in @QFWAF he proof of this theorem will e done in two stepsX

IF sup{Ψ (α,w) (f * )/(α, w) ∈ Λ} ≤ 1 2 F PF ∀f, ∃(α, w) ∈ Λ : Ψ (α,w) (f ) ≥ 1 2 F
First step: fy diret pplition of the rölder9s inequlity we hveX

Ψ (α,w) (f * ) = 1 0 {α(s)w(s)} 1/3 ds 3 = 1 0 α(s) 1/3 w(s) 2/3 ds 3 ≤ 1 0 α(s)ds 1 0 w(s)ds 2 ≤ 1 2 .
Second step: vet f e n ritrry positive densityF ke α

* ≡ 1 nd w * ≡ 2 D then (α * , w * ) ∈ Λ ndX Ψ (α * ,w * ) (f ) = 1 0 α * (s)w * (s) f 2 (s) ds = 1 2 1 0 1 f 2 (s)
ds ≥ 1 2 , sineD using the rölder9s inequlity we hveX

1 = 1 0 f 2/3 (s) 1 f 2 (s) 1/3 ds ≤ 1 0 f (s) ds 2/3 1 0 1 f 2 (s) ds 1/3 = 1 0 1 f 2 (s) ds 1/3
. his ompletes the proof of heorem QFQFPF

(t x,i ) -E(Y (t x,i )) = ε(t x,i )D we hveD s done y primn nd érez srirren @IWWIAD √ m ĝtrap n,m (x) -E ĝtrap n,m (x) = 1 √ m m j=1 1 2n N Tn -1 i=1 ϕ x,h f ε j (t x,i ) + ϕ x,h f ε j (t x,i+1 ) = 1 √ m m j=1 1 2n N Tn -1 i=1 ϕ x,h f (t x,i ) ε j (t x,i ) -ε j (x) + 1 √ m m j=1 1 2n N Tn -1 i=1 ϕ x,h f (t x,i+1 ) ε j (t x,i+1 ) -ε j (x) + 1 2n N Tn -1 i=1 ϕ x,h f (t x,i ) + ϕ x,h f (t x,i+1 ) 1 √ m m j=1 ε j (x) .
(3.67) e strt y ontrolling the lst term of this lst equtionF ell tht iqution @QFPWA yields for some t *

x,i ∈]t x,i , t x,i+1 [ tht 1 n = (t x,i+1 -t x,i )f (t * x,i )F prom the iemnn integrility of ϕ x,h nd f nd vemm QFSFI we otinD 1 2n N Tn -1 i=1 ϕ x,h f (t x,i ) + ϕ x,h f (t x,i+1 ) = 1 2 N Tn -1 i=1 ϕ x,h f (t x,i ) + ϕ x,h f (t x,i+1 ) f (t * x,i )(t x,i+1 -t x,i ) -→ m,n→∞ 1 -1 K(t) dt = 1. where d x,i = t x,i+1 -t x,i nd t * x,i ∈]t x,i , t x,i+1 [F he gentrl vimit heorem for iFiFdF vriles yieldsD 1 √ m m j=1 ε j (x) D -→ m→∞ Z where Z ∼ N (0, R(x, x)).
e shll prove now tht the two (rst terms of iqution @QFTUA tend to H in proility s n, m tends to in(nityF e will only study the (rst termD the seond one is treted nlogouslyF vetD

A m,n (x) = 1 √ m m j=1 1 2n N Tn -1 i=1 ϕ x,h f (t x,i ) ε j (t x,i ) -ε j (x) ∆ = 1 √ m m j=1
T n,j (x).

prom the gheyshev inequlityD it su0es to prove tht lim n,m→∞

E(A 2 m,n (x)) = 0F e hve for j = lD E(ε j (x)ε l (y)) = 0 so E(T n,j (x)T n,l (x)) = 0F reneD E(A 2 m,n (x)) = 1 m m j=1 m l=1 E(T n,j (x)T n,l (x)) = 1 m m j=1 E(T 2 n,j (x)).

IPT

Chapter 3

e hveD

E(T 2 n,j (x)) = 1 4n 2 N Tn -1 i=1 N Tn -1 k=1 ϕ x,h f (t x,i ) ϕ x,h f (t x,k )E ε j (t x,i ) -ε j (x) ε j (t x,k ) -ε j (x) = 1 4n 2 N Tn -1 i=1 N Tn -1 k=1 ϕ x,h f (t x,i ) ϕ x,h f (t x,k ) R(t x,i , t x,k ) -R(t x,i , x) -R(x, t x,k ) + R(x, x) .
ine E((T 2 n,j (x)) does not depend on j we getD

E(A 2 m,n (x)) = 1 4n 2 N Tn -1 i=1 N Tn -1 k=1 ϕ x,h f (t x,i ) ϕ x,h f (t x,k ) R(t x,i , t x,k ) -R(t x,i , x) -R(x, t x,k ) + R(x, x) ∆ = 1 4 B n,1 (x) -B n,2 (x) -B n,3 (x) + B n,4 (x) .
@QFTVA e otin using iqution @QFPWA for t *

x,i ∈]t x,i , t x,i+1 [D B n,1 (x) = N Tn -1 i=1 N Tn -1 k=1 f (t * x,i )f (t * x,k ) ϕ x,h f (t x,i ) ϕ x,h f (t x,k )R(t x,i , t x,k )d x,i d x,k .
he use of vemm QFSFI twie yieldsD

B n,1 (x) = N Tn -1 i=1 f (t * x,i ) ϕ x,h f (t x,i )d x,i x+h x-h ϕ x,h (t)R(t x,i , t) dt + O( 1 nh ) = x+h x-h ϕ x,h (t) N Tn -1 i=1 f (t * x,i ) ϕ x,h f (t x,i )R(t x,i , t)d x,i dt + O( 1 nh ) = x+h x-h x+h x-h ϕ x,h (s)ϕ x,h (t)R(s, t) ds dt + O( 1 nh ) = σ 2 x,h + O( 1 nh ).
sing @QFTRA we otinD

B n,1 (x) = R(x, x) - 1 2 α(x)C K h + o(h) + O( 1 nh ).
where

C K = 1 -1 1 -1 |u -v|K(u)K(v)dudv. ine lim n→∞ h = 0 nd lim n→∞ nh = ∞F husD lim n→∞ B n,1 (x) = R(x, x). @QFTWA 3.5. PROOFS
IPU gonsider now the term B n,2 (x)F e otin using vemm QFSFI twieD

B n,2 (x) = x+h x-h x+h x-h ϕ x,h (s)ϕ x,h (t)R(s, x) ds dt + O( 1 nh ) = x+h x-h ϕ x,h (s)R(s, x) ds + O( 1 nh ) = 1 -1 K(s)R(x -hs, x) ds + O( 1 nh ) = 0 -1 K(s)R(x -hs, x) ds + 1 0 K(s)R(x -hs, x) ds + O( 1 nh ). por s ∈] -1, 0[D ylor expnsion of R(•, x) round x yieldsD R(s, x) = R(x -sh, x) -shR (1,0) (x+, x) + o(h). imilrly for s ∈]0, 1[ we otinD R(x -sh, x) = R(x, x) -shR (1,0) (x-, x) + o(h). husD B n,2 (x) = R(x, x) -hR (1,0) (x+, x) 0 -1 s K(s) ds -hR (1,0) (x-, x) 1 0 s K(s) ds + o(h) + O( 1 nh ). reneD lim n→∞ B n,2 (x) = R(x, x). @QFUHA imilrlyD lim n→∞ B n,3 (x) = R(x, x). @QFUIA st is esy to see thtD lim n→∞ B n,4 (x) = lim n→∞ R(x, x) N Tn -1 i=1 N Tn -1 k=1 ϕ x,h f (t x,i ) ϕ x,h f (t x,k )) = R(x, x) 1 -1 K(t) dt 2 = R(x, x). @QFUPA
snserting @QFTWAD @QFUHAD @QFUIA nd @QFUPA in @QFTVA yieldsD

lim n,m→∞ E(A 2 m,n (x)) = 0.
his onludes the proof of heorem QFQFQF IPV Chapter 3 Appendix Lemma 3.5.1 (Integral approximation of a sum) Let u and v be two Lipschitz functions on [x -h, x + h], i.e, there exists two positive numbers l 1 and l 2 such that,

|u(s) -u(t)| ≤ l 1 |s -t|, |v(s) -v(t)| ≤ l 2 |s -t|. Let t x,1 < • • • < t x,N Tn be points in [x -h, x + h] and put d x,i = t x,i+1 -t x,i . Then, N Tn -1 i=1 u(t x,i )v(t x,i )d x,i = x+h x-h u(t)v(t) dt + ∆ n,h , for any t x,i ∈ [t x,i , t x,i+1
] for all i = 1, • • • , n and for some appropriate positive constants c 1 , c 2 and c 3 ,

|∆ n,h | ≤ c 1 l 1 h n sup t∈[0,1] |v(t)| + c 2 l 2 h n sup t∈[0,1] |u(t)| + 2 c 3 n sup t∈[x-h,t x,1 ] ∪[t x,N Tn ,x+h] |v(t)u(t)|.
Proof of Lemma 3.5.1. sn ftD let ∆

x,h = A -B whereD A = N Tn -1 i=1 u(t x,i )v(t x,i )d x,i nd B = x+h x-h u(t)v(t) dt. e hveD B = N Tn -1 i=1 t x,i+1 t x,i u(t)v(t) dt + t x,1 x-h u(t)v(t) dt + x+h t x,N Tn u(t)v(t) dt ∆ = B 1 + B 2 , where B 2 = t x,1 x-h u(t)v(t) dt + x+h t x,N Tn u(t)v(t) dtF yn the one hndD sine (t x,1 -(x -h)) ≤ sup 1≤i≤n d x,i nd (x + h -t x,N Tn ) ≤ sup 1≤i≤n d x,i we hveD |B 2 | ≤ 2c 3 sup t∈[x-h,t x,1 ] ∪[t x,N Tn ,x+h] |v(t)u(t)| sup 1≤i≤n d x,i .
yn the other hndD we hveD

A -B 1 = N Tn -1 i=1 t x,i+1 t x,i u(t x,i )v(t x,i ) -u(t)v(t) dt = N Tn -1 i=1 v(t x,i ) t x,i+1 t x,i u(t x,i ) -u(t) dt + N Tn -1 i=1 t x,i+1 t x,i u(t) v(t x,i ) -v(t) dt.
ine u nd v re vipshitz ontinuous we otinD

|A -B 1 | ≤ N Tn sup t∈[0,1] |v(t)|l 1 sup 1≤i≤n d 2 x,i + N Tn sup t∈[0,1] |u(t)|l 2 sup 1≤i≤n d 2 x,i . 3.5. PROOFS IPW ine nh ≥ 1D vemm QFPFI yields tht sup 1≤i≤n d x,i = O( 1 n ) nd N Tn = O(nh)F reneD |∆ n,h | = |A -B| ≤ |A -B 1 | + |B 2 | ≤ c 1 l 1 h n sup t∈[0,1] |v(t)| + c 2 l 2 h n sup t∈[0,1] |u(t)| + 2 c 3 n sup t∈[x-h,t x,1 ] ∪[t x,N Tn ,x+h] |v(t)u(t)|.
his onludes the proof of vemm QFSFIF Chapter 4

Optimal design for the nonparametric regression estimation applied to pharmacokinetics problems

Abstract: everl prolems of phrmokinetis re investigtedF he onentrtionEtime urve estimtion is onsidered using nonprmetri kernel estimtor through simultion study nd rel dt nlysisF por the ere nder the onentrtion gurve @egAD we introdue new kernel estimtor nd we showD through simultion studyD tht it outperforms the lssil trpeE zoidl estimtor in term of the estimtion errorF he prolem of estimting the iovilility is lso onsideredF he ruil prolem of (nding the optiml smpling design for the eg esE timtion is investigted using the qenerl imulted enneling lgorithmF he digoxin plsm onentrtion is used in the simultion studiesD for oth orrelted nd unorrelted oservE tionsF Key words: Pharmacokinetics, concentration curve, nonparametric estimation, AUC, bioavailability, optimal sampling design, General Simulated Annealing Algorithm.

Résumé: lusieurs prolèmes de phrmoinétique sont tritésF v9estimtion de l fontion de onentrtionEtemps est onsidérée en utilisnt un estimteur à noyuD à trvers une étude de simultion insi qu9une nlyse des données réellesF our l9estimtion de l9eire ous l goure de onentrtion @egAD nous introduisons un nouvel estimteur à noyu et nous montronsD à trvers une étude de simultionD que le nouvel estimteur est meilleur que l9estimteur trpezoidl lssique en terme de l9erreur d9estimtionF ve prolème de l9estimtion de l iodisponiilité est ussi onsidéréF ve prolème ruil de l9otention d9un pln d9éhntillonnge optiml pour l9estimtion de l9eg est ussi tritéD en utilisnt l9lgorithme de grdient onjugué ou l9lgorithme de reuit simulé générliséF v onentrtion plsmtique de digoxine est utilisée dns l9étude de simultionD pour donnés orrélées et non orréléesF Mots clés: Pharmacocinétique, courbe de concentration, estimation non paramétrique, AUC, biodisponibilité, plan d'échantillonnage optimal, algorithme de recuit simulé généralisé.

Introduction

ome of the most importnt prolems tht phrmokinetis reserhers re rought to invesE tigteD re the estimtion of the onentrtionEtime urve of some dministrted drug @or ny phrmologil gentA nd the ere nder this gurve @egAF his re represents the drug exposure of the orgnism over time nd is ritil in estimting the e0ieny of sorption @or iovililityAD sine the lter involves rtios of this reF sn this pperD the digoxin plsm onentrtion is onsidered in di'erent dministrtion wysF he onentrtion funtion is presented y sum of exponentil termsF sn prtiulrD following n orl dministrtionD the onentrtion is presented y three exponentil terms funtionD wheres fter intrvenous injetion of the drugD only two exponentil onentrtion urve is tkenF he numer of terms desries the numer of kinetilly homogeneous omprtments tht the drug invdes in the odyF he omprtmentl model we onsidered is lssil model tht mimi the dynmil proesses of sorptionD distriution nd elimintion of drug in the odyF sn phrmokinetisD the omprtments re usully di'erent tissue orgns within whih the onentrtion of drug is ssumed to e kinetilly homogeneousF por more detils on the omprtmentl models in phrmokinetisD we refer the reder to the work of hrgel et al.

@IWUHA nd qildi nd errier @IWUSAF e lso fousD in the simultion studyD on oth orrelted nd unorrelted oservtionsD sine in prtie when mesurements re tking from the sme experimentl units they re most likely to e orreltedF sn phrmokinetis studiesD fter the dministrtion of drug in the orgnismD lood smples re tken from sujets ording to spei( smpling plnF por instneD lood smples re tken every hlf n hour in period of T hoursD ut other smpling plns @or designsA mye more e0ient s it will e shown in this pper @setion RFSAF hen mesurements of the drug onentrtions from the lood smples re tkenD they re not the ext vlues of the rel onentrtion funtionD tht we re iming to estimteD ut rther some pprohed vluesF por instneD if we oserve the onentrtions otined from the sme lood smple in le RFID they re di'erent from n ssy to nother due to some mesurements errorsF por this resonD sttistiins dd residul omponent @lled errorA to the oserved onentrtion funtionF reneD we shll onsider the soElled (xed design regression model given yD

Y (t i,n ) = g(t i,n ) + ε(t i,n ) for i = 1, • • • , n @RFIA
where Y (t i,n ) is the oserved drug onentrtion t time t i,n D g is the unknown onentrtion funtionD the smpling design times (t i,n ) i re suh tht t 1,n < • • • < t n,n nd ε is n error proessF sn this hpterD severl prolems of phrmokinetis re investigtedF pirstD the estimtion of the onentrtion urve g in wodel @RFIA is onsideredF ientists often use prmetri methods to estimte the onentrtion urveD see for instne qildi nd errier @IWUSA for more detilsF roweverD these methods suppose tht the onentrtion urve hs spei( shpe nd tht it depends on some prmeters to e estimtedF por the ses where no prior knowledge onerning g is possile so tht only the estimtion of g is of interestF henD we propose nonprmetri pproh to estimte the onentrtion urve g sed on the kernel estimtor given y qsser nd wüller @IWUWAF yther nonprmetri kernel estimtors ould e usedD see for instne gheng nd lin @IWVIA nd more reently fenelmdni et alF @PHIWD PHIWA F e proveD through simultion study nd y onsidering rel dt nlysisD the good ehviour of the proposed estimtorF he estimtion of the eg is lso investigtedD for thisD we propose new nonprmetri estimtor whih is otined y n integrtion of the nonprmetri kernel estimtor of gF his eg 4.2. ESTIMATION OF THE CONCENTRATION CURVE IQQ estimtor is shownD through simultion studyD to outperform the lssil estimtorD whih is sed on the trpezoidl rule for pproximting n integrlD in terms of the wen qured irror @wiAF he iovilility estimtion prolem is lso onsidered in this simultion studyF pinllyD we investigte the ruil prolem of (nding the optiml smpling design times for the eg estimtionF sn other wordsD the gol is to hoose the est smpling times with respet to some riterion in order to estimte e0iently the egF his prolem ws (rst investigted y gherno' @IWSQA nd fox nd vus @IWSWAF everl pprohes were proposed to otin the optiml smpling timesD to estimte the eg y the trpezoidl pproximtionD using di'erent ojetive funtions nd di'erent lgorithmsF e refer to the work of ghoi et al. @PHHUA for review nd omprison of these methodsF sn this workD we onsidered the wi s n ojetive funtionD sine in sttistis it is n importnt tool to study the performne of n estimtorF st mesuresD in vergeD the squred di'erene etween the estimted vlues nd wht we seek to estimteF hen minimizing it with respet to smpling designsD would led to etter estimteF por unorrelted oservtionsD the gonjugted qrdient elgorithm is used nd for orrelted oservtionsD the qenerl imulting enneling elgorithm is onsideredF he hoie of the ltest ws essentilly due to its ility to hndle very omplex nonEliner ojetive funtions with very lrge numer of lol optim his hpter is orgnized s followsF sn etion RFPD we propose the nonprmetri kernel estimtor of the onentrtion urve nd we studyD through simultion experiment nd rel dt nlysisD the performne of this estimtorF sn etion RFQD the estimtion of the eg is onsidered y proposing new kernel estimtorF e ondut simultion study to illustrte its good ehviour nd to ompre it to the lssil trpezoidl estimtorD for oth orrelted nd unorrelted oservtionsF sn etion RFRD we onsider the estimtion of the iovilility through simultion studyF sn setion RFSD we investigte the prolem of (nding the optiml smpling points using wi s the ojetive funtion nd under pproprite lgorithmsF pinllyD etion RFT presents some omments nd onlusion of this workF 4.2 Estimation of the concentration curve sn this setionD we onsider the estimtion of the digoxin plsm onentrtion over timeF e use oth simulted dt nd numeril dt set generted in phrmeutil study onduted y gner nd tes @IWUQAF o estimte the onentrtionD we use the nonprmetri kernel estimtor given y qsser nd wüller @IWUWAnd rrt nd herly @IWVTA for t ∈ [0, 1] s followsD pigure RFIX e trjetory of the yrnsteinEhlenek proessF his model ws used in severl other worksD see for instneD utz nd h9ergenio @IWVQAD iegorsh nd filer @IWVWA ndD more reentlyD y felouni nd fenhenni @PHIQAD in order to estimte the eg nd to otin n optiml smpling design using lssil estimtor sed on the trpezoidl ruleF sn our workD we still onsider the sme model for the sme purposeD ut in nother ontextF sn ftD in this hpter we use nonprmetri regression pproh for estimting the onentrtion urve whih does not require spei( form of this urveD wheres in the previous worksD spei( prmetri models were imposed to the onentrtion funtionF sn our simultion studyD the error proess hs n utoovrine funtionX R(s, t) = Cov(ε(s), ε(t)) = σ(s)σ(t)ρ(s, t), @RFRA where ρ is the utoorreltion funtion @spei(ed lterA nd σ is the heterosedsti stndrd devition givenD for regression funtion gD yX σ(t) = 0.05 + 0.1g(t).

g GM n,h (t) = n i=1 Y (t i,n ) m i,n m i-1,n ϕ t,h (s) ds, @RFPA where ϕ t,h (s) = 1 h K t-s h D K is kernel of support [-1, 1]D h = h(n, m) is ndwidth with 0 < h < 1 nd the midpoints m i re given yX m 0,n = 0, m n,n = 1 nd m i,n = (t i + t i+1 )/2F
e onsider oth unorrelted errors generted y n error proess with the orreltion funtionX

ρ(s, t) =      1 if s = t, 0 if s = t, @RFSA
nd orrelted errorsD generted y the yrnsteinEhlenek error proessD with the orreltion funtionX ρ(s, t) = e -λ|s-t| . @RFTA sn the sequel we tke λ = 1F e visuliztion of this yrnsteinEhlenek error proess is displyed in pigure RFIF e (xed the smpling points numer n = 13 nd onsidered the onventionl smpling design (t i ) 1≤i≤n given y gner nd eyres @IWUUA s followsX 0, 0.25, 0.5, 0.75, 1, 1. 5,3,5,12,24,48,72,96. @RFUA 4.2. ESTIMATION OF THE CONCENTRATION CURVE IQS pigure RFPX he oservtions re in lk rossesD the estimtor is in dshed red line nd the onentrtion urve @RFQA is in plin lineF sn this simultion studyD we onsidered the men of IH vetors of generted oservtions following wodel @RFIA where g is given y @RFQA nd ρ is given y @RFSA or @RFTAF fsed on the qsser nd wüller estimtor g GM n,h of g oral D the optiml ndwidth is otined using the onjugteEgrdient lgorithms @gqeA @see plether nd eeves @IWRTAAD where the ojetive funtion is the sntegrted wen qured irror @swiA given yD given y @RFPA with the qurti kernel de(ned oveF he ndE width we hoose in our nlysis is otined using the ross vlidtion riterion @see for instne rärdle @IWWHAAD sine this method is wellEknown in nonprmetri regression prolemsF he results given in pigure RFQ on(rm those presented previously in our simultion studyD in other wordsD the estimtor g GM n,h trks very fithfully the dtF Remark 4.2.1 We note here that, in our simulation study, the function g oral has been adjusted to use samples in [0, 1] as follows:

IMSE GM n (h) = 1 0 E g oral (t) -g GM n,h (t) 2 dt = 1 0 n i=1 n j=1 σ(t i,n )σ(t j,n )ρ(t i,n , t j,n ) m i,n m i-1,n m j,n m j-1,n ϕ x,h (s)ϕ x,h (u) ds du + n i=1 g(t i,n ) -g(t) m i,n m i-1,n ϕ x,h (
goral (t) = -2.4e -10t×96 -2e -0.65t×96 -0.4e -0.0146t×96 .
In addition, in order to consider the boundary eect, that is to reduce the bias E(g ( n, h) GM (t))g oral (t) near the boundaries

[0, h] ∩ [1 -h, 1],
Hart and Wherly (1986) modied the estimator, so that it becomes a weighted average of the observations in the whole interval [0, 1], instead of the reduced interval [h, 1 -h]. This modication is done by taking the following kernel at the edges: where T is the investigtor9s lst smpling timeF por more detils on the use of prtil or totl @from 0 to ∞A res under the onentrtion urveD one n refer to the work of vovering et al.

K(t) = K(t)
@IWUSA nd gner nd eyres @IWUUAF o estimte this reD sientists often use qudrture methodsD see for instne the work of utz nd h9ergenio @IWVQAD felouni nd fenhenni @PHIQAF heir estimtor is sed on the trpezoidl numeril rule nd is given yX

AUC n (g) = 1 2 n-1 i=1 Y (t i,n ) + Y (t i+1,n ) (t i+1,n -t i,n ), @RFWA
en intuitive estimtor of eg is the integrl of the estimtor of the onentrtion urveF reneD we propose new kernel estimtor for eg(g)D whih is otined y integrting the estimtor g GM n,h D given y @RFPAD s followsX

AUC GM n,h (g) = n i=1 Y (t i,n ) 1 0 m i,n m i-1,n
ϕ t,h (s) dsdt, @RFIHA where ϕ t,h nd the midpoints m i,n re de(ned in etion PF sn order to ompre the proposed estimtor @RFIHA to the trpezoidl estimtor @RFWAD we onsider simulted oservtions from two di'erent onentrtion funtionsF he (rst one represents the digoxin plsm onentrtion fter n orl injetion given y g oral in @RFQAD nd the seond one represents its onentrtion fter n intrvenous injetionD given y utz nd h9ergenio @IWVQAD yX g int (t) = 3.117e -0.65t + 0.6657e -0.0146t , @RFIIA where int stnds for intrvenousF e generte dt from these two onentrtion funtions @the men of IHH vetorsA with the utoovrine de(ned y @RFRAD for oth unorrelted @RFSA nd orrelted @RFTA errorsD nd the onventionl design @RFUAF vikewiseD we use the qurti kernel nd the ndwidth otined y the gqeD to estimte eg(g oral ) nd eg(g int ) over the time intervl [0, 96]F he results re displyed in les RFP nd RFQF st is shown from these numeril resultsD in terms of wiD tht we n otin smller errors @up to 3% less in our simultionA nd etter estimtions when using the proposed nonprmetri kernel estimtorD thn the lssil trpezoidl estimtorD for oth orrelted nd unorrelted oservtionsF IQV Chapter 4 le RFPX he wiD the expettion of eg(g oral ) estimtor + the stndrd devition SDD the optiml ndwidth nd the estimtion vlue under the orreltions @RFSA nd @RFTAY AUC(g oral ) = 23.49F

F abs = AUC c AUC i D i D c ,
where AUC c nd D c re respetively the exposure nd the dose of the mediment dministrted in some hosen wy @suh sX orlD oulrD retl • • • AD AUC i nd D i re the exposure nd the dose of the mediment dministrted intrvenouslyF e onduted simultion study to estimte the solute iovilility of the digoxin when the sme dose is dministrted in oth wysF e onsidered the sme simulted dt presented in the previous setionD iFeFD the funtion g oral de(ned y @RFQA to represent the onentrtion urve fter orl dministrtion nd g int de(ned y @RFIIA for intrvenous dministrtionF he nturl .

he resultsD given in le RFRD re onsequene of the eg estimtions presented in the preE vious setionD iFeFD etter estimtions of eg give 4slightly4 etter estimtion of iovililityF 4.5 Optimal design for AUC estimation sn this setionD we re interested in (nding the optiml smpling design with respet to the wi for estimting the eg(g oral )F he (rst nd lst smpling times were (xed to H nd WTD the gqe ws used to otin the optiml design when the errors re unorreltedF roweverD when the errors re orrelted ording to the utoovrine funtion @RFTAD the gqe ould not e pplied sine it requires the di'erentility of the ojetive funtionD we used insted the qenerlized imulted enneling elgorithm @qeeAD see ing et al. @PHIQAF his lgorithm is essentilly known for its ility to hndle very omplex nonEliner ojetive funtions with very lrge numer of optimF e used the optiml ndwidth for the qw estimtor otined in etion RFQF he results re presented in les RFS nd RFT nd in pigure RFRF les RFP nd RFS show thtD we n signi(ntly derese the wi up to 50% when using the optiml design insted of the onventionl designD for instne when using the qsser nd wüller estimtor we n redue the MSE = 8.70 for the onventionl design to MSE = 4.29 using the optiml designD whih orresponds to derese of SH7F pigure RFR nd le RFT show tht most of the optiml smpling design points re loted in the elimintion phse nd fewer re in the sorption phse nd ner the pekD in ontrry to the onventionl smpling design presented y @RFUAF 4.6 Conclusion sn this workD nonprmetri regression method ws pplied to phrmokinetis prolemsF sn prtiulrD we were interested in three mjor prolemsX the nonprmetri estimtion of the onentrtion urveD of its re nd the derivtion of the optiml smpling designF pirstD we used IRH Chapter 4 le RFSX he wiD the expettion of eg(g oral ) estimtor + the stndrd devition SDD using the optiml design nd the estimtion vlue under the orreltions @RFSA nd @RFTAY AUC(g oral ) = 23.49F

IRI

pigure RFRX gomprison of the optiml designs nd the onventionl design for the estimtion of eg@g oral AF the nonprmetri kernel estimtor to estimte the onentrtion urveD without the knowledge of its shpeD in ontrry to the prmetri lssil methodsF e showedD through simultion studies nd rel dt nlysisD the good ehviour of the nonprmetri kernel estimtorF e proposedD thenD new kernel estimtor for the re under the onentrtion urveD whih is onstruted y integrting the onentrtion estimtorF e showed y simultion studyD tht the proposed estimtor outperforms the lssil eg estimtor sed on the trpezoidl ruleD in the sense tht it hs smller wiF pinllyD we investigted the prolem of (nding the optiml smpling design in the sense of minE imizing the wi with respet to ll smpling design of size nF he generted optiml design enled us to derese signi(ntly up to SH7 the error of estimtion @wiAD then the lssil onventionl designF woreoverD it is shown in our simultion study tht there is no signi(nt e'et of orrelted oservtions @through the uto ovrine funtionA on the performne of our estimtions in the phrmokineti prolems tht we investigtedF es noted y utz nd h9ergenio @IWVQAD the proposed smpling pointsD in ontrry to the onE ventionl oneD requires reltively lrge numer of smpling times in the elimintion phse nd does not tke into onsidertion the sorption phse nd onentrtion pekF roweverD s oserved y felouni nd fenhenni @PHIQAD the seletion of n optiml design method ould e djustedD y (xing ertin smpling times in the sorption phse nd round the pek nd then otin the remining optiml smpling timesF ACKNOWLEDGEMENT: his work ws developed in the frmework of qrenole elpes ht snstituteD supported y the prenh xtionl eserh egeny under the 4snvestissements d9venir4 progrm @exEISEshiEHPAF Chapter 5

Conclusion and perspectives sn this thesisD we were interested in the nonprmetri estimtion of the regression funtionD in the se where the error proess is of generl utoovrine funtionD this inludes the sttionry nd nonsttionry proessesF he model we onsidered is the (xedEdesign regression model with repeted mesurementsD where m experimentl units re eing oserved eh on n smpling pointsF e egn y onsidering well known kernel regression estimtorD proposed y qsser nd wüllerF e studied its symptoti ehvior when the numer of experimentl units nd the numer of oservtions tend to in(nityF e foused our studies in the se of the regulr design sequene where the smpling points re generted using spei( design densityF e then proposed two new kernel estimtors for the regression funtionF he (rst one is lled the projetion estimtorD it ws onstruted through projetion property nd using the eproduing uernel rilert pes propoertiesF he seond one is nmed the trpezoidl estimtorD onstruted using numeril ruleF e studied their symptoti ehviors when n nd m tend to in(nityD proved their symptoti normlity nd derived their optiml symptoti ndwidthsF e lso otined the optiml regulr smpling design for the trpezoidl estimtorF e then onduted simultion studies to test their performnes in (nite smple setsD nd to ompre them to the lssil qsser nd wüller estimtorF yur simultions on(rmed our theoretil resultsF pinllyD we onsidered n pplition to phrmokinetisD where we proposed the use of the nonprmetri regression estimtors to estimte the onentrtion funtion of given drugF e proposed new kernel estimtors to estimte the ere nder the onentrtion gurve @egA nd we showed its good performnes vi simultion study nd rel dt nlysisF e lso onsider the prolem of otining the optiml smpling design for the eg estimtion using the qenerl imulting enneling lgorithmF huring the preprtion of this thesisD some issues nd questions pperedD leding to some future reserhD tht we develop herefterF Models with dependent experimental units sn our thesis workD we onsidered the kernel regression modelD where the errors {ε j , j = 1, • • • , m} re iFiFdF proesses with the sme distriution s enteredD seond order proess εF he independene of the experimentl units is relisti ssumptionD for instneD in the longitudinl dt when oserving the heights of hildren over the yersF roweverD if we onsider sitution IRQ IRR Chapter 5 Conclusion and perspectives where we mesure some vrile on group of studentsD the students who were in prtiulr lsses in prtiulr shools tend to e more similr nd hene dependentF por thisD it is lso interesting to onsider the kernel regression model with repetedD dependent oservtions nd generlize our studiesF Dierent sampling times for each experimental units huring this thesisD we onsidered the nonprmetri regression model with repeted mesureE mentsD where the oservtions re mde on the sme smpling times (t i,n ) i=1,••• ,n for eh exE perimentl unitF his is the se when hoosing the smpling times is possileD for instneD to estimte the onentrtionEtime urveD the sientists (x the smpling time prior to the experiE mentF roweverD imgine set of pnel ount dtD where the experimentl units re oserved ontinuously nd for eh sujetD only the numer of ourrenes of the event re known t (nite distint oservtion time pointsF rene the times my vry from n experimentl unit to notherF un et al. @PHHUA onsidered this type of dt nd studied the prolem of estimting the prmeters of regression funtionF xún£ ezEentón et al. @IWWIA developed threeEstge pproh to estimte the regression funtion for suh type of dtF sn the nonprmetri seD one possile extension of the thesis work is to onsider the following nonprmetri regression modelX Y j (t i,j ) = g(t i,j ) + ε j (t i,j ) for j = 1, • • • , m nd i = 1, • • • , n j , where {ε j , j = 1, • • • , m} re iFiFdF with the sme distriution s enteredD seond ordered error proesseF en intuitive kernel estimtor one n use to estimte the regression funtion g is givenD for x ∈ [0, 1] yX ĝn,m (x) = m j=1 n j i=1 W i,j (x)Y j (t i,j ), where W i,j re some preised weightsF yne n use the qsser nd wüllerD the projetion or the trpezoidl weightsF st is interesting to study the symptoti ehvior of this estimtorD when the numer of experimentl units m nd the numers of oservtions n j for j = 1, • • • , m tend to in(nityF Data based bandwidth selection methods for correlated errors he dt sed ndwidth seletion methods re widely developed for unorrelted oservtionsF st hs een shown tht mny of themD inluding the well know rossEvlidtionD rekdown when deling with orrelted oservtionD see for instne eltmn @IWWHAD ghiu @IWVWA nd rrt @IWWID IWWRAF ypsomer et al. @PHHIA explined the prtil onsequenes of the sensitivity in the presene of orreltionF e very interesting prolem to improve the performne of the kernel estimtorD is thn to propose seletion method to otin the optiml ndwidth in the presene of orreltionD whih is the topi of our urrent projet 4yptiml trend estimtion under errors with wtérnE type eutoovrine nd pplitions to environmentl dt4D funded y qrenole elpes ht snstituteF IRS Estimating the change points for a non-dierentiable regression function e lssil hypothesis tht we mde on the nonprmetri regression modelD is tht the regresE sion funtion is twie di'erentileF his ws usefulD espeilly when studying the symptoti performnes of the kernel estimtorsF his ssumptionD s we hve seen y mny exmplesD is relisti for mny type of dtD suh sX the onentrtionEtime urve nd the heights of hildrenF ometimesD even smooth funtion my ontin disontinuityD or wht is lled hnge pointF por instneD go @IWUVA onsidered the nnul volume of the xile river from IVUIEIWUHD where there ws n rupt hnge in the rinfll tivity ner the endF re suggested tht the hnge ws ourred in the yer of IVWVF wüller @IWWPA proposed new method to estimte the hnge point while onsidering tht the oservtions re iFiFdF wny other uthors hve foused on estimting the hnge point for unorrelted errorD nd hene the prolem for orrelted oservtions is still n open questionF Nonparametric regression estimation for more regular autocovariance function sn our workD we ssumed tht the error proess is seond order proessD with nonEdi'erentile ovrine funtionF his is the seD for instneD for the iener proess nd the yrnsteinE hlenek proessF he work n e rought to other proesses with more regulr utoovriE ne funtionsD suh s the iterted frownin motionD whih ws onsidered y fenhenni et al.

@PHIQA to study the e'et of the regulrity on the qsser nd wüller estimtorF Bibliography I eltmnD xF F @IWWHA uernel moothing of ht with gorrelted irrorsF emerin ttistil essoitionD VSF URWEUSWF P ememiyD F @IWVSA edvned ionometrisF rrvrd niversity ressD gmridgeD weF Q endersonD FF @IWTHAF ome tohsti roess wodels por sntelligene test soresF wthE emtil wethods in oil ienesF tnford niversityF ressD tnfordD geD PHSEPPHF R eronszjnD xF @IWRRAF v heorie des xoyux eproduisnts et ses epplitionsF wtheE mtil roeedings of the gmridge hilosophil oietyD QWF IQQEISQF S ezzliniD eF @IWVRAF istimtion nd rypothesis esting por golletions of eutoregressive ime eriesF fiometrikD UIF VSEWHF T filerD eFtFD iegorshD FF @IWWHAF istimting integrls using qudrture methods with nd pplition in phrmokinetisF fiometrisD RTF IPHIEIPIIF U felouniD wFD fenhenni uF @PHISAF yptiml nd roust designs for estimting the onentrE tion urve nd the egF ndinvin tournl of ttistisX heory nd epplitionF RP RSQERUHF V fenedettiD tF @IWUUAF yn the xonprmetri istimtion of the egression puntionF tournl of the oyl ttistil oietyD QWF PRVEPSQF W fenelmdniD hFD fenhenniD uFD vouhihiD F @PHIWAF rpezoidl rule nd smpling desings for the nonprmetri estimtion of the regression funtion in models with orrelted errorsF https://arxiv.org/abs/1806.04896

IH fenelmdniD hFD fenhenniD uFD vouhihiD F @PHIWAF he reproduing kernel rilert spe pproh in nonprmetri regression prolems with orrelted oservtionsF ennls of the snstitute of ttistil wthemtisF II fenhenniD uFD gmnisD F @IWWPAF mpling hesigns for istimting sntegrls of tohsti roessesF he ennls of ttistisD PHF ITIEIWR F IP fenhenniD uFD hdiD wF @PHHUAF xonprmetri istimtion of everge qrowth gurve with qenerl xonsttionry irror roessF gommunitions in ttistisX heory nd wethodsD QTF IIQUEIIVTF IQ fenhenniD uFD hdiD wFD uD F @PHIQAF he e'et of the regulrity of the error proess on the performne of the kernel regression estimtorsF wetrikD UTF UTSEUVIF IRU IRV BIBLIOGRAPHY IR fikelD FtFD rerzerg eFwF @IWUWA oustness of design ginst utoorreltion in timeD sF esymptoti theoryD optimlity for lotion nd liner regressionF ennls od sttistisD UF UUEWSF IS fiedermnnD FD hetteD rF @PHHIA winimx optiml design for nonprmetri regressionX further optimlity property of the uniform distriutionF wyhe TX edvnes in wodelE yriented hesign nd enlysisF @etkinsonD eFgFD rklD FD wüllerD FqFDedsAF hysiE erlgD reidelergF IQEPHF IT flnkeD hFD fosqD hF @PHHVAF egression istimtion nd redition in gontinuous imeF tournl of the tpn ttistil oiety @xihon ôkei qkki uihôAD QVF ISEPTF IU foxD qFiFFD vusD rFvF @IWSWAF hesign of ixperiments in non liner situtionsF fiometrikD RTF UUEWHF IV ghengD uFpFD vinD FiF @IWVIAF xonprmetri istimtion of the egression puntionF F hrsheinlihkeitstheorie verwF qeieteD SUF PPQEPQQF IW gherno'D rF @IWSQAF volly optimum designs for estimting prmetersF ennls of wtheE mtil ttistisD PRF SVTETHPF PH goD qFF @IWUVAF he prolem of the xileX gonditionl solution to hnge point proE lemF fiometrikD TSF PRQEPSIF PI F F ghiu @IWVWAF fndwidth eletion for uernel istimtion with gorrelted xoiseF ttisE tis nd roility vettersD VF QRUEQSRF PP ghoiD vFD g'oD fFD ohdeD gF @PHHUAF yptiml smpling times in ioequivlene studies using simulted nneling lgorithmF ttistis nd gomputingD IUF QQUEQRU PQ hvidinD wD qiltinnD hFwF @IWWSAF xonliner models for repeted mesurement dtF ghpmn nd rllF PR hetteD rFD epelyshevD eFD higljvsky eF @PHITAF yptiml designs in regression with orreE lted errorsF he ennls of ttistisD RRF IIQEISPF PS hetteD rFD horningD uFD uonstntinou wFD @PHIUAF yptiml designs for ompring regression models with orrelted oservtionsF gomputtionl ttistis nd ht enlysisF http: //dx.doi.org/10.1016/j.csda.2016.06.017F

PT hidiD FD nd vouniD hF @PHIQAF esymptoti esults for the egression puntion istimte on gontinuous ime ttionry nd irgodi htF tournl of ttistis nd isk wodellingD QIF IPWEISHF PU hu'ullD FfFD etoutD FD wentréD pF @PHHPAF he use of simulted nneling for (tting optiml popultion designsF gomputer wethods nd rogrms in fiomediineD TWF PSEQSF PV ifromovihD F @PHHVAF yptiml sequentil design in ontrolled nonprmetri regressionF ndinvin tournl of ttistisD QSF PPTEPVSF PW pnD tF @IWWPAF hesignEdptive nonprmetri regressionF tournl of the emerin ttisE til essoitionD VUF WWVEIHHRF SV oD gFF @IWTSAF he theory of lest squres when the prmeters re stohsti nd its pplition to the nlysis of growth urvesF fiometrikD SPF RRUERSVF SW oD gFF @IWTTAF govrine djustment nd relted prolems in multivrite nlysisF wultivrite nlysisF edemi pressD xew orkDVUEIHQF TH ksD tFD lviskerD hF @IWTTAF hesigns por egression rolems ith gorrelted irrorsF he ennls of wthemtil ttistisF QUD TTEVWF TI ksD tFD lviskerD hF @IWTVAF hesigns por egression rolems ith gorrelted irrorsX mny prmetersF he ennls of wthemtil ttistisF QWD TWEUWF TP ksD tFD lviskerD hF @IWUHAF hesigns por egression rolems ith gorrelted irrors sssF he ennls of wthemtil ttistisF RID PHSUEPHURF TQ hwrtzD vF @IWTRAF ous ispe rilertiens d9ispe etoriels opologiques et xoyux essoiés @xoyux eproduisntA F tournl d9enlyse wthémtiquesD IQF IISEPSTF ISI TR hrgelD vF uEongD F endrewD fFgF @IWUHAF epplied fiophrmeutis nd hrmokiE netisD sixth editionF TS uD FD gmnisD F @IWWQAF mpling hesigns for istimtion of ndom roessF tohsE ti roesses nd their epplitionsD RTF RUEVWF TT unD tFD ongD FD reD F @PHHUAF egression nlysis of pnel ount dt with dependent oservtion timesF tohsti roesses nd their epplitionsD TQF IHSQEIHSWF TU gnerD tFqFD tesD tFhF @IWUQAF iquivlene vk in higoxin lsm vevelsF he tournl of the emerin wedil essoition PD IWWEPHRF TV gnerD tFqFD eyresD tFF @IWUUAF fiovilility ssessmentX wethods to estimte totl re nd totl mount exreted nd importne of lood nd urine smpling sheme with pplition to digoxinF tournl of phrmokinetis nd iophrmeutisD SF SQQESSUF TW ingD F quin D uomel fD roengD tF @PHIQAF qenerlized imulted enneling for e0ient glol optimiztionX the qene kge for F he tournlD SF IQEPVF UH immermn h vD xúñezEentón D il frmi rF @IWWVAF gomputtionl spets of likelihoodE sed estimtion of (rstEorder ntedependen modelsD tF ttistF gompututF imulD THF TUEVRF UI hoD FD o F @PHIPAF equentil design for nonprmetri infereneF he gndin tournl of ttistisD RHF QTPEQUUF UP higljvskyD eF hetteD rF epelyshevD eF @PHIHAF e new pproh to optiml design for liner models with orrelted oservtionsF tournl of the emerin ttistil essoitionD IHSF IHWQEIIHQF

  < +∞. sn dditionD we prove the symptoti normlity of the proposed estimtorD see heorem PFRFQF e lso give theoretil omprison etween the new estimtor ĝpro n nd qsser nd wüller9s estimtor ĝGM n,h D proving tht ĝpro n hs n symptotilly smller vrine thn ĝGM n,h D wheres they re oth symptotilly unisedF he ft tht the projetion estimtor hs smller symptoti vrine n e rgued yD in the one hndD the qsser nd wüller9s estimtor ĝGM n,h does not tke into ount the orreltion requirementF yn the other hndD the estimtor ĝGM n,h

  2.3. CONSTRUCTION OF THE ESTIMATOR USING THE RKHS APPROACHRQ r de(nition of P |Tn D we hve for ny g ∈ V Tn P |Tn f -f, g = 0.

  eA gives the following lemm onerning the jump funtion αF Lemma 2.3.1 If Assumption (A) is satised then the jump function α is a positive function.

  presents the symptoti normlity of the projetion estimtor @PFUA for ny error proess εF Theorem 2.4.3 Suppose that the assumptions of Theorem 2.4.1 are satised. Moreover assume that n m = O(1) as n, m → ∞, that lim n,m→∞ nh 2 = ∞ and that lim n,m→∞

  por the error proessD we shll onsider oth the iener of utoovrine funtion R(s, t) = min(s, t)D nd the yrnsteinEhlenek proess with utoovrine R(s, t) = e -|s-t| F he weight wD hosen hereD is the uniform density on [0, 1]D iFeFD w ≡ 1 on [0, 1]D we onsider the optiml ndwidth with respet to the ext IMSE of the two estimtorsD iFeFD inf 0<h<1

  (rst term of the right side of @QFTSAF ine Y

  )du, for t ∈ [0, h] ∩ [1 -h, 1].

  l9estimteur de projetion présente une vrine symptotique plus file peut être rgumenté pr deux pointsF remièrementD l9estimteur de qsser et wüller ĝGM n,h ne tient ps ompte de l orréltion des oservtionsF heuxièmementD l9estimteur ĝGM n,h est une pproximtion d9une intégrle et que l meilleure pproximtion linéire d9une intégrle est sée sur une propriété de projetionD voir pr exemple fenhenni et gmnis @IWWPAF in(nD nous menons une étude de simultion (n d9étudier les performnes de l9estimteur proposé ĝpro n dns un ensemle (ni d9éhntillonngeD où nous démontrons ses onnes perforE mnes pour les éhntillons de petite tilleF insuiteD nous le omprons ve l9estimteur de qsser et wüller ĝGM n,h D pour di'érentes vleurs du nomre d9unités expérimentles m et di'érentes vleurs de l tille de l9éhntillonnge nF gette simultion on(rme nos résultts théoriquesF

	mh 3 n,m < +∞.
	in outreD nous prouvons l normlité symptotique de l9estimteur proposéD voir le héorème
	PFRFQF xous e'etuons églement une omprison théorique entre le nouvel estimteur ĝpro

n et l9estimteur lssique de qsser et wüller ĝGM n,h F xous montrons que l vrine de ĝpro n est plus petite que elle de ĝGM n,h D lors qu9ils sont tous les deux symptotiquement non iisésF ve fit que

  RWProposition 2.4.3 Suppose that Assumptions (A) and (B) are satised. Moreover, assume that

	1 h sup
		).	@PFIRA
	he following proposition ontrols the residul vrine	σ 2 x,h

m -Var ĝpro n (x)F 1≤i≤n d i ≤ 1 and let,

  Suppose that Assumptions (A) -(D) are satised. Moreover assume that (T n ) n≥1 is a regular sequence of designs generated by a density function f (see Denition 2.4.1). If lim

	SH			Chapter 2
	Proposition 2.4.6
	and,
	lim n→∞	mn 2 h	σ 2 x,h m	-Var ĝpro

n (x) ≥ C . @PFIUA nder the stronger ssumption (D) on the kernel K nd using regulr sequene of designs @see he(nition PFRFIAD we otin the symptoti expression of the vrine s shown y the following propositionF n→∞ h = 0 and lim n→∞

  Theorem 2.5.1 Suppose that Assumptions (A), (B) and (D) are satised. Moreover assume that (T n ) n≥1 is a regular sequence of designs generated by a density function f (see Denition 2.4.1). If lim

n→∞ h = 0 and lim n→∞

  10 -4 4.689 × 10 -3 4.778 × 10 -3 HFIRP le PFPX he integrted squred isD integrted vrineD swi nd the optiml ndwidth for n = 10 nd di'erent vlues of m under the yrnsteinEhlenek error proessD for the qw nd the projetion estimtorsF

	10 m GM IH P ro GM SH P ro GM IHH P ro GM IH P ro GM SH P ro GM IHH P ro 0.897 × n = 10 m Ibias 2 1.508 × 10 -3 4.507 × 10 -2 4.658 × 10 -2 HFQQS Ivar IMSE h opt 1.304 × 10 -3 4.399 × 10 -2 4.530 × 10 -2 HFQPI 2.662 × 10 -4 9.494 × 10 -3 9.760 × 10 -3 HFIWV 1.981 × 10 -4 9.228 × 10 -3 9.426 × 10 -3 HFIVU 1.505 × 10 -4 4.826 × 10 -3 4.977 × 10 -3 HFISR Ibias 2 Ivar IMSE h opt 2.596 × 10 -3 8.821 × 10 -2 9.080 × 10 -2 HFQVU 2.494 × 10 -3 8.703 × 10 -2 8.952 × 10 -2 HFQVT 4.481 × 10 -4 1.848 × 10 -2 1.893 × 10 -2 HFPQT 4.097 × 10 -4 1.822 × 10 -2 1.863 × 10 -2 HFPQU 2.299 × 10 -4 9.390 × 10 -3 9.620 × 10 -3 HFIVT 1.885 × 10 -4 9.265 × 10 -3 9.453 × 10 -3 HFIVU 2.7 Proofs

sn this setionD we shll omit the index n in t i,n when there is no miguityF Proof of Lemma 2.2.1 he(neD for suitle prtition

  n .Second case. gonsider now the se where t x,1 < x-h nd t x,N Tn > x+hF por i = 2, . . . , N Tn -2

	2.7. PROOFS			US
	roposition PFRFQ is then proved for the (rst seF
	setD
	tx,i+1	tx,2	x+h
	γ x,i =	ϕ x,h (t) dt, γ x,1 =	ϕ x,h (t) dt, γ x,N Tn -1 =	ϕ x,h (t) dt and γ x,N Tn = 0.
	tx,i	x-h	t x,N Tn -1

  2 HFRHV IHU le QFUX he swi nd the redutions in the swi of ĝtrap n using the uniform designD theoretil optiml design nd estimted optiml design when R(s, t) = σ 2 ρ |s λ -t λ |/λ nd n = 5. m T rap unif T rap opt rIM SE λ T rap opt rIM SE λ λ using the uniform designD theoretil optiml design nd estimted optiml design when R(s, t) = σ 2 ρ |s λ -t λ |/λ nd n = 10. m T rap unif T rap opt rIM SE λ T rap opt rIM SE λ λ using the uniform designD theoretil optiml design nd estimted optiml design when R(s, t) = σ 2 ρ |s λ -t λ |/λ nd n = 20. m T rap unif T rap opt rIM SE λ T rap opt rIM SE λ λ using the uniform designD theoE retil optiml design nd estimted optiml design when R(s, t) = σ 2 ρ |s λ -t λ |/λ nd n = 30. m T rap unif T rap opt rIM SE λ T rap opt rIM SE λ λ

	IHV						Chapter 3
	le QFIHX he swi nd the redutions in the swi of ĝtrap
	S S	HFQTTI HFITVP	HFQIQV HFIRVV	IRFPV7 IIFST7	HFQITU HFIRQR	IQFSH7 IRFUV7	SFIS RFRT
	IH IH	HFQSQU HFIPHI	HFPWVV HFIHST	ISFSR7 IPFHW7	HFPWWP HFHWUQ	ISFRI7 IWFHQ7	RFHW RFVT
	PH PH	HFQRUS HFHWTI	HFPWIP HFHVRH	ITFPH7 IPFSU7	HFPWPV HFHVTI	ISFUR7 IHFR7	RFRH QFTW
	QH QH	HFQRSR HFHVVI	HFPVVU HFHUTV	ITFRP7 IPFUV7	HFPVRR HFUSVT	IUFTU7 IQFVV7	QFRS RFIR
	3.5 Proofs
	Proof of Lemma 3.2.1.
	le QFVX he swi nd the redutions in the swi of ĝtrap
	S	HFIWTW	HFIUUI	IHFHT7	HFIVPP	UFSH7	SFHT
	IH	HFITUR	HFIRWR	IHFUW7	HFIRVU	IIFIW7	QFWI
	PH	HFISPU	HFIQSS	IIFPT7	HFIQHS	IRFSR 7 QFPI
	QH	HFIRUU	HFIQHW	IIFRQ7	HFIQRT	VFVU7	RFSH
	le QFWX he swi nd the redutions in the swi of ĝtrap
	S	HFITWW	HFIRVU	IPFSP7	HFIRSU	IRFPT7	RFQS
	IH	HFIPUR	HFIHWT	IPFIR7	HFIIHT	IIFQR7	QFVP
	PH	HFIHPP	HFHWHI	IIFVT7	HFHVVS	IQFQW7	RFQR
	QH	HFHWRU	HFHVQT	IIFUQ7	HFHVQW	IIFQI7	QFWH

n n n

  s) ds

	2 here the E(X) stnds for the expettion of rndom vrile XF pigure RFP shows tht the dt, dshed urve whih represents the estimtorD pprohes well the plin urve tht represents the rel onentrtion urve to e estimtedF his visulistion shows tht our nonprmetri pproh gives good estimtor for the onentrtion urveF 4.2.2 Real data analysis sn this setionD we present our nlysis of dt set onsidered y gner nd tes @IWUQAF he dt re digoxin plsm onentrtions fter n orl dministrtion of tretment whih 1 1.5 3 5 12 24 48 72 96 assay 1 0 0.03 0.39 0.65 0.74 0.79 0.6 0.38 0.28 0.21 0.13 0.09 0.08 assay 2 0 0.02 0.43 0.64 • • • 0.71 0.55 0.37 0.27 0.20 0.15 0.09 0.04 onsists of 0.Hours 0 0.25 0.5 0.75 e use the estimtor g GM n,h

25mg tlets of digoxinF he mesurements were tken following hosen time line s presented in le RFIF por one sujetD duplite ssys were run on eh plsm smple nd we took the verge of the two ssys to present the onentrtionsF IQT Chapter 4 pigure RFQX he oservtions in le RFI re in irles nd the estimtor g GM n,h is in dshed lineF le RFIX higoxin plsm onentrtion @mgGmlA in sujet dministered orllyF

  wiD the expettion of eg(g int ) estimtor + the stndrd devition SDD the optiml ndwidth nd the estimtion vlue under the orreltions @RFSA nd @RFTAY AUC(g int ) = 39.17F n = 13istimtor MSE E( ÂUC) + SD yptiml h istimtion

	n = 13	istimtor MSE E( ÂUC) + SD yptiml h istimtion
	norrelted	AUC n	VFTS	PQFUV + PFWQ	E	PQFUW
	oservtions	AUC	GM n,h	VFTH	PQFSU + PFWQ	HFHWP	PQFSU
	gorrelted	AUC n	VFUU	PQFUV + PFWS	E	PQFUI
	oservtions	AUC	GM n,h	VFUH	PQFSR + PFWS	HFHWV	PQFRU
	le RFQX he norrelted	AUC n	IPFTS	QWFVI + QFSH	E	QWFVS
	oservtions	AUC	GM n,h	IPFPT	QWFPH + QFSH	HFHTR	QWFPQ
	gorrelted	AUC n	IPFWP	QWFVI + QFSR	E	QWFWQ
	oservtions	AUC	GM n,h	IPFRV	QWFIS + QFSQ	HFHUI	QWFPT
	4.4 Estimation of the bioavailability
	he solute iovilility is de(ned y the following formulX

  4.5. OPTIMAL DESIGN FOR AUC ESTIMATIONIQW le RFRX he estimted iovilility @the rel vlue of F abs is 0.600AF

		n = 13	istimtor istimtion
	norrelted		Fn	HFSWU
	oservtions		F GM n,h	HFTHI
	gorrelted		Fn	HFSWR
	oservtions		F GM n,h	HFSWV
	estimtions we used reX
	Fn =	AUC n (g oral ) AUC n (g int )	,	F GM n,h =	AUC AUC GM n,h (g oral ) GM n,h (g int )

  IRW QH prwy tFtF @IWWHAF equntil hesign for the nonprmetri regression of urves nd surE fesF gomputer iene nd ttistisX roeedings of the PPnd ennul ymposium on the snterfeD wihign tte niversityD wsF IHREIIHF QI perreirD iFD xú nezEentónD FD odríguezEóoD tF @IWWUAF uernel egression istimtes of qrowth gurves sing xonsttionry gorrelted irrorsF ttistis 8 roility vettersD QRF RIQERPQF QP pletherD F eevesD gFwF @IWRTAF puntion minimiztion y onjugte grdientsF he gomE puter tournlD UF IRWEISRF QQ primnD FD érez srirrenD qF @IWWIAF xonprmetri egression in wodels with ek irror9s trutureF tournl of wultivrite enlysisD QUF IVHEIWTF QR qsserD FD wüllerD rFqF @IWUWAF uernel istimtion of egression puntionsF veture xotes in wthemtisD USUF PQETVF QS qsserD FD wüllerD rEqF @IWVRAF istimting egression puntion end heir herivtives fy he uernel wethodF ndinvin tournl of ttistisD IIF IUIEIVSF QT qsserD FD wüllerD rFqF uöhlerD FD wolinriD vF nd rderD eF @IWVRAF xoprmetri regression nlysis of growth urvesF he nnls of sttistis IPD PIHEPPWF QU qildiD wFD errierD hF @IWUSAF hrmokinetisF hrugs nd hrmeutil sienesF eE ond editionF QV qhoshD wFD qrizzleF tFiFD enD Fu @IWUQAF xonprmetri methods in longitudinl studiesF emerin ttistil essoitionD TVF PWEQTF QW qrizzleD tFiFD ellenF hFwF @IWTWAF enlysis of growth nd dose response urvesF fiometris PSD QSUEQVIF RH rärdle F @IWWHAF epplied xonprmetri egressionF ionometri oiety wonogrphs IWF RI rrtD tFhFD herlyF FiF @IWVTAF uernel egression istimtion sing epeted wesureE ments htF tournl of the emerin ttistil essoition VID IHVHEIHVVF RP rrtD tFhF @IWWIAF uernel egression istimtion with ime eries irrorsF oyl ttistil oiety fF SQ IUQEIVUF RQ rrt tDhF@IWWRAF eutomted uernel moothing of hependent ht y sing ime eries gross lidtionF oyl ttistil oiety fF ST SPWESRPF RR utz hFD h9ergenioF @IWVQAF ixperimentl design for estimting integrls y numeril qudrture with pplitions to phrmokinet studiesF fiometrisD QWID TPIETPVF RS vovering iFqFD wqilvery sFtFD wwilln sFD ostowryk F @IWUSAF gomprtive iovilE ilities from trunted lood level urvesF tournl of hrmeutil ienesD TRD ISPIE ISPRF RT wüllerD qFrF @IWVRAF yptiml hesigns por xonprmetri uernel egressionF ttistis end roility vetters PD PVSEPWHF RU wüllerD qFrF @IWWIAF mooth yptimum uernel istimtors xer indpoints F fiometrikF UVDQD SPIESQHF RV wüllerD qFrF @IWWPAF ghngeEpoints in nonprmetri regression nlysisF he ennls of ttistisF PHF UQUEUTIF RW xúñezEentón F @IWWUAF vongitudinl dt nlysisX xonEsttionry error strutures nd ntedependent modelsF epplied tohsti wodels nd ht enlysisD IQF PUWEPVUF SH xúñezEentónD F odriquzEóoD tFwFD ieuD F @IWWWAF vongitudinl ht with nonsttionE ry errorsX nonprmetri threeEstge pprohF oiedd de istdísti e snvestigión ypertivF est VF PHIEPQIF SI xúñezEentónD F oodworthD qF @IWWRAF enlysis of vongitudinl ht with neqully ped yservtions nd imeEhependent gorrelted irrorsF fsywisgD SHF RRSERSTF SP ypsomerD tFD ngD FD ngD F @PHHIAF xonprmetri egression with orrelted errorsF ttistil ieneD ITF IQREISQF SQ rzenD iF @IWSWAF ttistil snferene on ime eries y rilert pe wethodsF heprtE ment of ttistisF tnford niversityF ehnil eportF PQF SR iegorsh FFD filer eFtF @IWVWAF yptiml design llotions for estimting re under urves for studies employing destrutive smplingF tournl of hrmokinetis nd fioE phrmeutisD IUD RPQESHUF SS ottho'D FpFD oyD FxF @IWTRAF e generlized multivrite nlysis of vrine model useful espeilly for growth urve prolemsF fiometrikD SIF QIQEQPTF ST riestlyD wFfFD ghoD wFF @IWUPAF xonprmetri funtion (ttingF tournl of oyl ttisE til oietyD QRF QVREQWPF SU msyD tFyFD ilvermnD fFF @PHHSAF puntionl ht enlysisF pringer eries in ttisE tisF

Acknowledgments