Keywords: Statistique non paramétrique, Estimation récursive, Environnement, Valvométrie Nonparametric statistic, Recursive estimation, Environment, Valvometry Application and case studies, mathematical statistics, nonparametric methods, smoothing and nonparametric regression

La protection de l'environnement, en particulier celle des systèmes aquatiques, est une des priorités de nos sociétés. L'utilisation de capteurs biologiques permettant de tester la qualité de l'eau en continue est une voie possible de surveillance intégrée des milieux aquatiques. Cette démarche a été mise en place avec succès sur des mollusques bivalves équipés d'électrodes légères qui respectent leur comportement naturel, on parle alors de valvométrie.

Le but de cette thèse est de calculer et traiter automatiquement la vitesse de mouvement des valves de mollusques bivalves installés dans divers milieux aquatiques. Les années d'enregistrements déjà acquises nous permettrons, à partir de nos modèles, de détecter s'il existe des variations de la vitesse de mouvement des valves liées aux variations de température.

Plus particulièrement, nous avons étudié les dérivées de différents estimateurs non paramétriques d'une fonction de régression : l'estimateur récursif de Nadaraya-Watson, l'estimateur de Johnston, l'estimateur de Wand-Jones ainsi que l'estimateur de Révész. Nous avons aussi pris en compte la version déterministe de l'estimateur de Nadaraya-Watson.

Pour chacun des estimateurs nous avons mené une étude sur les comportement asymptotiques en particulier la convergence presque sûre et la normalité asymptotique.

Nous avons illustré numériquement ces propriétés et appliqué ces nouvelles méthodes d'estimations sur des données réelles afin de valider, ou non, les hypothèses environnementales émises par les biologistes.

Je tiens aussi à remercier toutes les personnes qui m'auront accompagné au cours de ces quatre ans. Merci tout d'abord à Emmanuelle pour m'avoir soufflé l'idée de faire une Licence de Maths, je t'avais dit que tu serais la première sur la liste de mes remerciements donc si jamais un jour, en tapant mon nom sur Google tu tombes sur ce manuscrit : Voilà ! Merci à tous les anciens de Jussieu qui m'auront accompagné le long de ces trois ans : à Morgane pour son accueil à Manille et ces vacances philippines plus que bienvenues avant de démarrer la rédaction d'un article, à Nina, ma soeur de clinique, pour toutes ces escapades à Belleville, à Noémie pour sa présence en général et ses séjours à Bordeaux illuminés par sa petite princesse, à Antoine et Anne-Lise (qui je l'espère me fera dédicacer un exemplaire de manuscrit), à Mehdi et Dhafer qui malgré la distance restent les membres d'une équipe dont on ne peut pas écrire le nom dans un manuscrit, Pierre pour cet enterrement de vie de garçon qui aura repoussé la soumission du susnommé article de deux jours pour cause de mal de crâne (Encore félicitations Casanova) et merci à Mathilde pour avoir lancé ma carrière dans le mannequinat.

Un grand grand merci à Léa pour ces discussions variées et hilarantes, je suis trop content de te compter à nouveau parmi mes amies. Merci à Thomas de l'ENS de Lyon pour les discussions cinéma, tennis et Sergio Ramos, à Thomas de l'ENS de Lyon, petit ange arrivé trop tard, pour tous ses traquenards constructifs, le confit de canard et à Margot pour lui emplir le coeur de joie.

Merci à Vasil¨e¨i¨o¨s le plus beau de l'IMB, désolé que tes luttes aient moins bien convergées que mes estimateurs et merci à Athina pour tous ses soleils qui ont égayés mon tableau.

Ces soleils ont aussi profité à mes cobureaux que je remercie pour avoir supporté mes soupirs. Yann merci pour la carte Rabiot qui deviendra forcément collector et Baptiste, chaton, c'était vraiment un plaisir de t'avoir face à moi pendant cette trop courte année.

Merci à Lara pour les gateaux, les cours de rattrapage d'arabe, Alexandre à cause de qui je balance du vamos a graillar tous les midis, Antonin qui, je pense, serait jaune s'il n'apparaissait pas dans cette liste (comme Corentin) et Marcu Antone, champion du monde contre son gré, qui je le sais, attends plus le derby contre l'Etoile Filante Bastiaise de dimanche que cette soutenance. Merci aussi à Thibaut (et Cannelle) pour les five, Inch'Allah Metz remonte en Ligue 1 Conforama et Strasbourg redescend en Domino's Ligue 2. Ah et au fait, Roxane, aujourd'hui je veux une médaille ! Merci à Jonathan et Nicolas pour l'accompagnement mutuel au cours de nos multiples AVC, cancers, tumeurs et crises cardiaques, Merci à Nikola pour les rakias, les barbecues, les soirées, les "bouge !", les canapés. T'es le meilleur des présidents ! Hvala za sve. Bilo je zadovoljstvo raditi ovu doktorsku tezu u isto vreme kad i ti. Merci aux serbes pour avoir ambiancé ces trois ans, en particulier Stefan, Saša, Diane et Jelena que je remercie encore plus pour la traduction.

Merci à Alice qui entre l'Appolo, le Cri de la Mouette, la Piscine ou le Sodexo de Limours aura été la meilleure collègue dont on puisse rêver. J'en profite pour remercier Dr Léonard qui, grâce à sa prescription d'antihistaminiques, m'a permis d'écrire ce manuscrit dans de bonnes conditions.

Et enfin, merci aux joueurs de squash à Philippe the cream of the cream, le meilleur basque de tout l'univers et la personne la plus généreuse de tout les temps. Wallah j'an rajoute pas ! Et bravo Carole pour nous avoir supporté pendant les parties de PES un peu tard dans la nuit. Et merci à Elsa avec qui j'ai partagé les moment d'euphorie comme les grosses déprimes de Dubrovnik à Guadara...Guadajala... Guadalaraja... bon au Mexique quoi. C'était un bonheur de faire cette thèse avec vous ! Je souhaiterai aussi remercier Lana del Rey, Nina Kraviz, Kendrick Lamar, les frères Andrieu, Migos et Vivaldi pour m'avoir accompagné tout au long de mes périodes de codage et de rédaction. Merci aux éditeurs de Football Manager et à Richairo Zivkovic, Quincy Promes et Memphis Depay pour m'avoir fait réaliser que dans la vie il n'y a pas seulement des articles à rédiger ou des cours à préparer, il y a surtout une coupe du monde 2022 à gagner à l'arrachée contre l'Autriche (3-2 a.p.). Merci à Edinson Cavani pour maintenir la flamme PSG encore allumée. Edi si jamais un jour, en tapant mon nom sur Google, tu tombes sur ce manuscrit : cette thèse t'es dédiée Matador.

Parce qu'il n'y a que la miff que la miff, merci à la famille aussi, oncles, tantes, cousins.

Comme le 94 c'est le Barça, j'ai une pensée particulière pour Lamine (là c'est Thauvin qui vient de marquer un but de fou furieux, je vais arrêter de remercier les Marseillais sinon ça va mal finir pour Guingamp), Ryad qui malgré ses piètres qualités à FIFA m'aura apporté beaucoup de fous rires au cours de ces trois ans (gobelet), à Amir mon frère et à Amina pour tout leur soutien.

Merci à ma marraine pour ses conseils avisés et à mon parrain qui m'aura fait monter l'Aubisque, mon plus bel accomplissement jusqu'à aujourd'hui (thèse comprise).

Merci à Karine pour avoir partagé avec moi la fin de cette aventure et et qui partagera aussi le début de la nouvelle. Même si tu aimes Orelsan et que tu manges des cookies comme repas tu es à l'ammoniaque.

Merci à mon père pour ses toujours judicieux conseils, que ce soit pour résoudre les problèmes de chapitrage, pour ma carrière ou pour la vie en général. Merci à mon frère qui doit sûrement s'attendre à ce que je le vanne mais non, je t'aime frère ! Merci à ma mère, pour tout. Cette introduction est constituée de trois parties. La première partie est consacrée au domaine d'application de cette thèse : la valvométrie. La deuxième est dédiée à la présentation des différents estimateurs non-paramétriques étudiés au cours de cette thèse. Les principaux résultats obtenus dans la thèse sont énoncés dans la dernière partie.

Table des matières

Valvométrie

L'objectif de cette thèse est de développer de nouvelles méthodes d'estimation statistiques avec application en valvométrie. Cette section aura pour but d'introduire ce domaine d'études et notamment d'en souligner l'intérêt dans la surveillance de la pollution des eaux côtières.

Enjeux environnementaux

Il est aujourd'hui évident que la protection de l'environnement est une priorité de nos sociétés. La surveillance des systèmes aquatiques en particulier est un enjeu majeur, les océans ayant un rôle primordial dans la régulation du climat. En effet, la photosynthèse du phytoplancton, des micro-organismes végétaux, émet entre 50 et 85% de l'oxygène consommé par les êtres vivants tout en captant entre 20 et 35% du dioxyde de carbone produit par l'activité humaine. Au même titre que les forêts, les milieux aquatiques peuvent donc être considérés comme l'un des poumons de la planète.

Si les dégradations environnementales engendrées en pleine mer ne sont pas négligeables, la pollution marine est en grande majorité d'origine terrestre et produites par des activités humaines, on parle de pollution terrigène. Cette forme de pollution trouve ses sources dans l'agriculture (utilisation massive d'engrais, de nitrates ou de pesticides), dans l'industrie (métaux lourds, hydrocarbures) ou par le mode de vie des habitants (déchets, eaux usées).

On estime qu'en 2035, près de 65% de la population mondiale vivra à moins de 100 kilomètres du littoral et pourrait ainsi être directement impactée par cette pollution, que ce soit par la consommation de produits issus de la mer ou par la destruction de la faune et flore littorale ( [15]). De plus, une fois que ces polluants on atteints l'océan, ils sont susceptibles, grâce aux courants marins, de circuler tout autour du globe et de contaminer l'ensemble des milieux aquatiques. La surveillance de la pollution des eaux côtières est donc doublement importante. D'une part pour évaluer la qualité de l'eau et d'autre part pour estimer la quantité de polluants susceptible de se diluer dans les océans.

Bioindicateurs

Il y a de nombreux paramètres à prendre en compte pour surveiller la qualité de l'eau : la concentration de produits chimiques, le pH, la température. l'utilisation de bioindicateurs est de plus en plus répandue. On peut citer par exemple les lichens qui sont utilisés pour déterminer la concentration de dioxyde de souffre [6], les abeilles pour détecter la pollution aux métaux lourds ( [4]) ou encore les coléoptères carnivores dont la population indique la présence ou l'absence d'insectes ravageurs ( [21]). Dans le cas de la pollution aquatique, l'utilisation des mollusques bivalves est particulièrement adaptée ( [10], [28], [2]).

Bivalves

Les bivalves sont une classe de mollusques dotés de deux coquilles (ou valves) s'articulant autour d'une charnière leur permettant de s'ouvrir et de se fermer. La L'étude du mouvement des mollusques bivalves, ou valvométrie, peut donc permettre d'obtenir de nombreuses informations sur l'état de l'animal et donc sur l'état de son environnement. L'observation de comportements anormaux peut ainsi servir de signal d'alarme sur l'éventuelle présence d'un contaminant ( [24], [13]). Ces animaux étant sédentaires, l'étude à long terme de leurs mouvements peut témoigner de l'évolution de la qualité de l'eau sur des années.

De nombreuses études ont déjà été effectuées, que ce soit sur les pontes ( [1]), la croissance ( [27]) ou les perturbations du rythme biologique ( [31]). Au cours de cette thèse, nous nous focaliserons sur les vitesses d'ouverture et de fermeture des valves, une vitesse exagérée des mouvements des valves pouvant être indicateur de stress et donc associée à une perturbation du système aquatique.

Valvométrie HFNI

Les premières études sur le comportement des bivalves remontent au début du 20 ème siècle [18]. Si au début les études étaient d'abord effectuées en plaçant une feuille de papier millimétré derrière un bocal transparent, de nombreuses méthodes ont été utilisées depuis. La Figure 1.2 représente un exemple de méthode utilisé pour mesurer l'effet de la salinité sur la palourde d'Afrique de l'Ouest ( [9]). Une des valve est fixée sur un socle avec du ciment, l'autre, laissée libre, est reliée à une jauge avec un hameçon et du fil métallique.

Aujourd'hui les systèmes ne reposent plus sur des jauges mécaniques mais sur des éléctro- Les données sont envoyées en utilisant soit le réseau de téléphonie mobile et un protocole de transfert de données GPRS (General Packet Radio Service), soit directement internet si une prise ethernet est disponible localement. Ces dispositifs permettent une autonomie complète du système. Les premières cartes filles immergées ont une durée de vie de 2 ans, les cartes filles à la surface sont alimentées avec des batteries à panneaux solaires.

On obtient ainsi à la fin de la journée un tableau de 54 000 lignes par bivalve. La Figure 1.8 illustre graphiquement le comportement d'une huître sur le site de Locmariaquer. On

Estimation non paramétrique d'une fonction de densité

Supposons que l'on dispose de n variables aléatoires indépendantes X 1 , ¨¨¨, X n , de même loi de densité g inconnue. Lorsque l'on ne possède aucune information sur cette densité, on se trouve dans un cadre d'estimation non paramétrique. L'objectif est donc de proposer un estimateur de g construit à partir des données X 1 , ¨¨¨, X n .

Histogramme

La méthode la plus simple et intuitive est l'histogramme. Pour cela, on considère que les données sont réparties sur un compact I " ra, bs que l'on découpe en m classes régulières I j " ri j , i j`1 r pour j " 1 ¨¨¨, m avec i 1 " a et i m`1 " b. Ainsi, pour chaque x P ra, bs, il existera un indice j tel que x P I j . Les intervalles étant de tailles égales, on peut définir h P R `tel que pour tout indice j, i j`1 ´ij " 2h. On définit alors l'estimateur p g n pxq pour tout x P ra, bs et pour tout n ě 1 par

p g n pxq " 1 2nh n ÿ k"1 1 X k PI j . (1.2)
Cette méthode présente cependant l'inconvénient de dépendre fortement du nombre de classes choisi. De plus, pour calculer l'estimateur en un point x, on pourrait prendre en compte des observations éloignées et en ignorer d'autre beaucoup plus proches du fait de la position de x dans l'intervalle I j . On peut résoudre ce dernier problème en considérant pour chaque x un intervalle centré en x dans lequel on prend les observations en compte.

On définit alors l'estimateur histogramme mobile p g n pxq pour tout x P R et pour tout

n ě 1, par p g n pxq " 1 2nh n ÿ k"1 1 X k Prx´h,x`hr .
(1.3)

On remarque qu'en notant K U la fonction définie, pour tout x P R, par

K U pxq " 1 2 1 |x|ď1 ,
on obtient ainsi l'estimateur de Rosenblatt ([25])

p g n pxq " 1 nh n ÿ k"1 K U ˆx ´Xk h ˙. ( 1
p g n pxq " 1 n n ÿ k"1 1 h k K ˆx ´Xk h k ˙.
(1.7)

Estimation non paramétrique d'une fonction de régression

On suppose maintenant que l'on dispose de n couples de variables aléatoires indépendants pX 1 , Y 1 q, ¨¨¨, pX n , Y n q suivant le modèle de régression (1.1). On considère que pX n q est une suite de variables indépendantes et identiquement distribuée de loi de densité g, connue ou non. L'objectif est d'estimer la fonction de régression f .

Régressogramme

De la même manière que l'histogramme (1.2), une façon de calculer p f n consiste à considérer que les variables aléatoires pX n q sont définies sur un compact ra, bs et à diviser ce compact en m classes régulières I 1 , ¨¨¨, I m de taille 2h. Ainsi, pour chaque x P ra, bs, il existera un indice j tel que x P I j . On définit alors p f n pxq pour tout x P ra, bs et pour tout n ě 1, par

p f n pxq " n ř k"1 Y k 1 X k PI j n ř k"1 1 X k PI j . ( 1
p f n pxq " n ř k"1 Y k K U ˆx ´Xk h ṅ ř k"1 K U ˆx ´Xk h ˙. (1.9)
Estimateurs à noyau de la fonction de régression

On peut généraliser (1.9) en optant pour un noyau K quelconque. De plus, en prenant là aussi en compte une taille de fenêtre h n dépendant du nombre d'observation,s on obtient l'estimateur de , [34]), défini pour tout x P R et pour tout n ě 1,

par f N W n pxq " n ř k"1 Y k K ˆx ´Xk h n ṅ ř k"1 K ˆx ´Xk h n ˙. (1.10)
Une version récursive de cet estimateur est définie ( [11]) pour tout x P R et pour tout

n ě 1, par p f n pxq " n ř k"1 Y k h k K ˆx ´Xk h k ṅ ř k"1 1 h k K ˆx ´Xk h k ˙. (1.11)
On reconnaît au dénominateur l'estimateur de Wolverton-Wagner p g n de la densité g. Ainsi en notant, pour tout x P R,

p h n pxq " 1 n n ÿ k"1 Y k h k K ˆx ´Xk h k ȯn obtient p f n pxq " p h n pxq p g n pxq .
(1.12)

Dans le cas où la densité g est connue, il n'est pas nécessaire de l'estimer. On peut alors utiliser une version simplifiée de l'estimateur récursif de Nadaraya-Watson, l'estimateur de Johnston ([16]), défini pour tout x P R tel que gpxq ą 0 et pour tout n ě 1, par

r f n pxq " 1 ngpxq n ÿ k"1 Y k h k K ˆx ´Xk h k ˙. (1.13)
Nous considérons aussi une version alternative de l'estimateur de Johnston, l'estimateur de Wand-Jones ( [33]) défini pour tout x P R et pour tout n ě 1 par

q f n pxq " p h n pxq gpxq " 1 n n ÿ k"1 Y k gpX k qh k K ˆx ´Xk h k ˙. ( 1
et q f 1 n pxq " 1 n n ÿ k"1 Y k gpX k qh 2 k K 1 ˆx ´Xn h n ˙. (1.20)
Le dérivée de l'estimateur de Révész (1.15) est quand à elle donnée, pour tout x P R, par

p f 1 n pxq " ˆ1 ´1 nh n K ˆx ´Xn h n ˙˙p f 1 n´1 pxq `Yn nh 2 n K 1 ˆx ´Xn h n ˙. (1.21)
Enfin, la dérivée de la version déterministe de l'estimateur de Nadaraya-Watson (1.17) est définie, pour tout x P r0, 1s par Ces résultats sont retrouvés à travers des simulations numériques. Nous y illustrons les convergences presque sûres ainsi que les normalités asymptotiques, en faisant ressortir l'influence de f 2 pxq pour les estimateurs r f 1 n pxq et q f 1 n pxq. Ce résultat peut être retrouvé dans la Figure 1.12 qui représente les histogrammes de N " 2 000 calculs de p f 1 n pxq aux points x " 0.5 et x " 0.9. Pour x " 0.5, nous avons f 2 pxq " 0 et pour x " 0.8, nous avons f 2 pxq " 21.65. Comme attendu, nous avons bien deux histogrammes similaires pour les deux points avec a dérivée de l'estimateur récursif de Nadaraya-Watson alors que pour la dérivée de l'estiateur de Johnston la variance est plus grande en x " 0.9.

p f 1 n pxq " 1 nh 2 n n ÿ k"1 Y k K 1 ˆx ´tk h n ˙. ( 1
Nous effectuons de plus une comparaison avec un estimateur par polynômes locaux ( [32]) pour illustrer l'apport de notre approche dans le cadre de données éparses. Ce résultat peut-être observé dans la Figure 1.13, où l'on observe le calcul de p f n pour des observations éparses sur les bords. On voit bien que l'estimateur récursif de Nadaraya-Watson est bien moins impacté par le manque de données que l'estimateur par polynômes locaux. Nous arrivons ainsi aux même conclusions qu'une étude portant sur la même période révélant une perturbation environnementale due à la suractivité enzymatique causée par une hausse de la température ( [12], [7]).

Article 2 : A nonparametric statistical procedure for the detection of marine pollution

Dans cet article, nous nous intéressons au comportement asymptotique de p f 1 n , la version déterministe de la dérivée de l'estimateur récursif de . Nous obtenons, pour tout x Ps0, 1r, la convergence presque sûre

lim nÑ`8 p f 1 n pxq " f 1 pxq p.s.
Un résultat similaire avait déjà été obtenu pour les noyaux à supports compacts. De plus il était nécessaire que le bruit pε n q possède un moment d'ordre ą 2 fini. Pour obtenir la convergence presque sûre avec un moment d'ordre 2 fini, il fallait ajouter la condition sur la fenêtre h n " 1{n α avec α ą 1{4. Ici, notre convergence presque sûre est assurée pour le noyau Gaussien ainsi que pour un bruit possédant un moment d'ordre 2 fini sans condition supplémentaire pour α.

Nous obtenons aussi les normalités asymptotiques

a nh 3 n ´p f 1 n pxq ´E " p f 1 n pxq ı¯L ÝÑ N `0, σ 2 ξ 2 ȇt pour 1{5 ă α ă 1{3, a nh 3 n ´p f 1 n pxq ´f 1 pxq ¯L ÝÑ N `0, σ 2 ξ 2 ˘.
Là encore, ce résultat avait été obtenu pour des noyaux à supports compacts. Ici, nous avons la normalité asymptotique pour un noyau Gaussien ce qui diminue considérablement la variance asymptotique ( [14]).

Dans la section consacrée à la simulation numérique, nous mettons en valeur l'apport du noyau Gaussien pour la variance asymptotique. Nous soulignons aussi le fait que la variance asymptotique ne dépend pas de x et qu'elle est constante entre 0 et 1. La Figure De plus, si le bruit pε n q possède un moment d'ordre ą 2 fini, on a

a nh 3 n ´p f 1 n pxq ´f 1 pxq ¯L ÝÑ N ˆ0, gpxqξ 2 σ 2 2gpxq ´p1 ´3αq ˙.
Une comparaison entre la variance asymptotique de la dérivée de l'estimateur de Révész et la dérivée de l'estimateur récursif de Nadaraya-Watson y est discutée. Cette comparaison est illustrée dans la partie simulation.

Chapitre 2

Nonparametric recursive estimation of the derivative of the regression function with application to sea shores water quality Abstract. This paper is devoted to the nonparametric estimation of the derivative of the regression function in a nonparametric regression model. We

Introduction

Environmental and water protection should be tackled as a top priority of our society. It is forecasted that in 2035, nearly 60% of the world's population will live within 65 miles of the sea front [19]. Water quality monitoring is therefore fundamental especially on the coastline. On the one hand, marine pollution comes mostly from land based sources.

On the other hand, this pollution can lead to the collapse of coastal ecosystems and cause public health issues. In this context, there is a critical need to develop a real-time reliable field assay to monitor the water quality within a decision making process. Among them, bioindicators are more and more commonly used. Endemic species are the most suitable bioindicators for the assessment of the quality of the coastal environment. For exemple, oysters, a well-known filter-feeding mollusc, feature a relevant sentinel organism to evaluate water quality. These animals being sedentary, they can witness the water quality evolution in a specific location.

The interest in investigating the bivalve's activities by recording the valve movements has been explored for water quality surveillance. This area of interest is known as valvometry.

The basic idea of valvometry is to use the bivalve's ability to close its shell when exposed to a contaminant as an alarm signal (e.g. [13], [26] and [35]). Thus, recording the shell gaping activity of oysters is an effective method to study their behavior when facing water pollution (e.g. [30] and [17]). Nowadays, valvometric techniques produce high-frequency data, enabling online and in situ studies of the behavior of bivalve molluscs. They allow autonomous long-term recordings of valve movements without interfering their normal behavior. The goal of this paper is to propose a nonparametric statistical procedure based on the estimation of the derivative of the regression function in order to evaluate the velocity of the valve opening/closing activity.

A wide range of literature is available on nonparametric estimation of a regression function. We refer the reader to ( [25], [11], [18], [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF]) for some excellent books on density and regression function estimation. Here, we shall focus our attention on the Nadaraya-Watson estimator of the regression function ( [24], [START_REF] Watson | Smooth regression analysis[END_REF]). The almost sure convergence of this estimator was established by [28], while its asymptotic normality was proven by [33].

Later, [8] proposed three data-sharpening versions of the Nadaraya-Watson estimator in order to reduce the asymptotic variance in the central limit theorem.

In this paper, we investigate an alternative approach, based on three recursive versions of the Nadaraya-Watson estimator (see [1], [3], [5], [12], [14], [20], [21], [START_REF] Wand | Kernel Smoothing[END_REF]). As it is well-known, recursive estimation procedures are specially useful when the observations are gathered sequentially. The three recursive versions of the Nadaraya-Watson estimator allow us to update the estimates as new observations are collected during the monitoring process, avoiding the need to recalculate a new estimate from the whole data. To the best of our knowledge, [27] is the only reference available on the derivative of the recursive Nadaraya-Watson estimator. However, the derivative is obtained by differentiation of the kernels, which is absolutely not necessary. In addition, some arguments are missing in the proof of the almost sure convergence and the explicit evaluation of the variance for the asymptotic normality is not provided. Consequently, our first goal is to carefully investigate the asymptotic behavior of the derivative of those three estimators. Our second goal is to illustrate our nonparametric estimation procedure on high-frequency valvometry data, in order to detect irregularities or abnormal behaviors of bivalves.

Another strategy for the recursive estimation of the regression function and of its derivatives is the local polynomial fitting approach, which was successfully implemented by [23], [START_REF] Vilar-Fernández | Recursive estimation of regression functions by local polynomial fitting[END_REF], [START_REF] Vilar-Fernández | Recursive local polynomial regression under dependence conditions[END_REF]. The advantages of local polynomial fitting are twofold. The estimates are easily computable and they have nice asymptotic properties. The main drawback of this approach is that it may be quantitatively affected by sparse regions of the random design [34], as it requires compactly supported kernels.

The paper is organized as follows. Section 2.2 deals with our nonparametric estimation procedure of the derivative of the regression function. We establish in Section 2.3 the pointwise almost sure convergence as well as the asymptotic normality of our estimates and we compare their asymptotic variances. Section 2.5 is devoted to a real data application on the survey of aquatic system using high-frequency valvometry. All the proofs of the nonparametric theoretical results are postponed to Appendices A and B.

Nonparametric estimation of the derivative

The relationship between the distance of two valves pY n q and the time of the measurement pX n q can be seen as a nonparametric regression model given, for all n ě 1, by

Y n " f pX n q `εn (2.1)
where pε n q are unknown random errors. In all the sequel, we assume that pX n q is a sequence of independent and identically distributed random variables with positive probability density function g. Our purpose is to estimate the derivative of the unknown regression function f which is directly associated with the velocity of the valve opening/closing activities of the oysters. For example, in an inhospitable environment, oysters behavior will be altered. Consequently, detecting changes of the closing and opening speed can provide insights about the health of oysters and so can be used as bioindicators of the water quality.

We recall that the Nadaraya-Watson estimator of the link function f is defined as

p f N W n pxq " n ř k"1 Y k K ˆx ´Xk h n ṅ ř k"1 K ˆx ´Xk h n ˙, (2.2)
where the kernel K is a chosen probability density function and the bandwidth ph n q is a sequence of positive real numbers decreasing to zero. In our situation, we focus our attention on the recursive version of the Nadaraya-Watson estimator [14] of f given, for any x P R, by

p f n pxq " n ř k"1 Y k h k K ˆx ´Xk h k ṅ ř k"1 1 h k K ˆx ´Xk h k ˙. (2.
3)

The denominator should, of course, be taken positive. It coincides with the recursive version of the Parzen-Rosenblatt estimator ( [29], [31]) of the probability density function g. For any x P R, denote

p h n pxq " 1 n n ÿ k"1 Y k h k K ˆx ´Xk h k ˙, p g n pxq " 1 n n ÿ k"1 1 h k K ˆx ´Xk h k ˙(2.4)
which can be recursively calculated as

p h n pxq " n ´1 n ĥn´1 pxq `Yn nh n K ˆx ´Xn h n ˙(2.5) and p g n pxq " n ´1 n ĝn´1 pxq `1 nh n K ˆx ´Xn h n ˙. (2.6)
This modification allows dynamic updating of the estimates. In the special case where g is known, a simplified version of the Nadaraya-Watson estimator of f , introduced by [21],

is given by

r f n pxq " p h n pxq gpxq .
(2.7)

In the same vein, an alternative estimator of f when g is known, was proposed by [START_REF] Wand | Kernel Smoothing[END_REF]. It is defined, for any x P R, by

q f n pxq " 1 n n ÿ k"1 Y k gpX k qh k K ˆx ´Xk h k ˙. (2.8)
The derivatives of p f n pxq, r f n pxq, and q f n pxq are given, for any x P R such that gpxq ą 0, by

p f 1 n pxq " p h 1 n pxq p g n pxq ´p h n pxqp g 1 n pxq p g 2 n pxq , (2.9) r f 1 n pxq " p h 1 n pxq gpxq ´p h n pxqg 1 pxq g 2 pxq , (2.10)
q f 1 n pxq " 1 n n ÿ k"1 Y k gpX k qh 2 k K 1 ˆx ´Xk h k ˙.
(2.11)

Theoretical results

In order to investigate the asymptotic behavior of these derivative estimates, it is necessary to introduce several classical assumptions. First of all, denote by F n the σ-algebra of the events occurring up to time n, F n " σpX 1 , ε 1 , . . . , X n , ε n q.

pA 1 q The kernel K is a positive symmetric bounded function, differentiable with bounded derivative, satisfying

ż R Kpxqdx " 1, ż R K 1 pxqdx " 0, ż R xK 1 pxqdx " ´1, ż R x 2 K 1 pxqdx " 0, ż R x 4 Kpxqdx ă 8, ż R x 4 |K 1 pxq|dx ă 8.
pA 2 q The regression function f and the density function g are bounded continuous, twice differentiable with bounded derivatives.

pA 3 q The driven noise pε n q is a martingale difference sequence satisfying, for all n ě 1, Erε n |F n´1 s " 0 and Erε 2 n |F n´1 s " σ 2 a.s. where σ 2 ą 0. Moreover, for all n ě 1, X n and ε n are conditionally independent given F n´1 .

On the one hand, it is not necessary to assume that the kernel K is compactly supported. On the other hand, we are not in the restrictive situation where the noise pε n q is a sequence of independent random variables. Our martingale assumption allows many general dependence structure of the random noise pε n q. It is also important to notice that we are able to avoid the strict stationarity assumption usually made on the distribution of pX n , Y n q. Finally, the bandwidth ph n q is a sequence of positive real numbers, decreasing to zero, such that nh n tends to infinity. For the sake of simplicity, we shall make use of

h n " 1{n α with 0 ă α ă 1.
Our first result on the almost sure convergence of our estimates is as follows.

Theorem 2.3.1. Assume that pA 1 q, pA 2 q and pA 3 q hold. Then, if 0 ă α ă 1{3, we have for any x P R such that gpxq ą 0, lim nÑ8 p f 1 n pxq " f 1 pxq a.s.

(2.12)

lim nÑ8 r f 1 n pxq " f 1 pxq a.s. (2.13) lim nÑ8 q f 1 n pxq " f 1 pxq a.s. (2.14)
Démonstration. The proof is given in 2.5.

Remark 2.3.1. Under additional assumptions, it should be possible to establish almost sure rates of convergence of the uniform deviation of p f 1 n pxq ´f 1 pxq for x lying in a given compact set. In the same vein, uniform strong law of the logarithm were previously established for the non recursive version of the Nadaraya-Watson estimator ( [6], [10], [22]).

Our second result is devoted to the asymptotic normality of our estimates. Denote

ξ 2 " ż R `K1 pxq ˘2dx.
(2.15)

Theorem 2.3.2. Assume that pA 1 q, pA 2 q and pA 3 q hold and that the noise pε n q has a finite conditional moment of order ą 2. Then, as soon as 1{5 ă α ă 1{3, we have for any

x P R such that gpxq ą 0, the pointwise asymptotic normality

a nh 3 n `p f 1 n pxq ´f 1 pxq ˘L ÝÑN ˆ0, ξ 2 p1 `3αqgpxq σ 2 ˙, (2.16
)

a nh 3 n `r f 1 n pxq ´f 1 pxq ˘L ÝÑN ˆ0, ξ 2 p1 `3αqgpxq `f 2 pxq `σ2 ˘˙, (2.17)
a nh 3 n `q f 1 n pxq ´f 1 pxq ˘L ÝÑN ˆ0, ξ 2 p1 `3αqgpxq `f 2 pxq `σ2 ˘˙. (2.18)
Démonstration. The proof is given in Appendix B.

Remark 2.3.2. One can realize that the derivate of the Nadaraya-Watson estimator p f 1 n pxq is more efficient that r f 1 n pxq and q f 1 n pxq as its asymptotic variance is the smallest one. The more f pxq is far away from 0, the more one should make use of p f 1 n pxq. Moreover, the smallest values of ξ 2 are given by kernels with non-compact support as the Gaussian kernel

Kpxq " 1 ? 2π exp ˆ´x 2 2 ˙, ξ 2 " 1 4 ?
π .

Remark 2.3.3. Denote by p f LP n pxq the recursive local polynomial estimator of f pxq. It follows from Corollary 1 in [START_REF] Vilar-Fernández | Recursive estimation of regression functions by local polynomial fitting[END_REF] with h n " 1{n α where 1{5 ă α ă 1{3 that

a nh 3 n `p p f LP n q 1 pxq ´f 1 pxq ˘L ÝÑN ˆ0, p1 ´2αq 2 ζ 2 p1 ´αqgpxq σ 2 ẇhere ζ 2 " ˆżR x 2 Kpxqdx ˙´2 ż R x 2 `Kpxq ˘2dx.
The main drawback of this approach is that it may be quantitatively affected by sparse regions of the random design pX n q, as it requires K to be compactly supported [34]. For example, if K stands for the Epanechnikov kernel, one can easily check that ζ 2 " 15{7 » 2.1429. However, we already saw that for the standard Gaussian kernel, ξ 2 " 1{p4 ? πq » 0.1410. Consequently, our approach offers much more flexibility in the choice of the kernel K, and the asymptotic variance for p f 1 n pxq is in general smaller than that of the derivative of p f LP n pxq.

Simulated data

This section is devoted to numerical experiments in order to evaluate the performances of our derivative estimates. The data are generated from the nonparametric regression model

Y n " f pX n q `εn , (2.19)
where the regression function f and its derivative f 1 are defined, for all x in r0, 1s, by f pxq " px `2q sinp4πx 2 q `2 sinp8πxq and f 1 pxq " sinp4πx 2 q `8πxpx `2q cosp4πx 2 q `16π cosp8πxq.

The source of variation pε n q is a sequence of independent and identically distributed random variables with Np0, 1q distribution. The random observation pX n q is a sequence of independent random variables sharing the same distribution which is a mixture of three uniform distribution

gpxq " p1 ´pq 2 g 1 pxq `pg 2 pxq `p1 ´pq 2 g 3 pxq (2.20)
where 1{2 ă p ď 1 and

g 1 pxq " 1 1 ´p 1 xPr0,1´ps , g 2 pxq " 1 2p
´1 1 xPr1´p,ps , g 3 pxq "

1 1 ´p 1 xPrp,1s .
One can observe that for p " 1, g coincides with the uniform distribution on r0, 1s. This distribution is introduced in order to illustrate the good performances of our statistical procedure in front of sparse regions of the random design pX n q. We implement our statistical procedure with a large sample size n " 10 000 since we have large datasets in the application described in Section 2.5.

We first illustrate the pointwise almost sure convergence of the three estimators p f 1 n pxq, r f 1 n pxq and q f 1 n pxq when the random observations pX n q are uniformly distributed over r0, 1s, which means that p " 1. The choice of the kernel K is not crucial for the pointwise almost sure convergence and we have chosen to make use of the Epanechnikov kernel. In order to select the parameter α of the bandwidth, we use the standard cross validation method. n pxq (dashed line), and q f 1 n pxq (dash-dotted line), to f 1 pxq (solid line).

In order to illustrate the pointwise asymptotic normality of our estimates, we implement a simulation study based on N " 2 000 realizations. We numerically check the asymptotic normality at points x " 0.5 and x " 0.8 for our three estimators. One can see in Figure 2.2 that the distributions of our three estimators are normally distributed and centered around 0. We observe the effect of f 2 pxq on the asymptotic variance of r f 1 n pxq and q f 1 n pxq. Indeed, for x " 0.5, we have f 2 pxq " 0, while for x " 0.8, we have f 2 pxq " 21.65 which explains the differences between the asymptotic variances. It can be also shown that the mean squared error (MSE) of p f 1 n pxq is much more smaller than the MSE of the non-recursive version of the Nadaraya-Watson estimator. In term of asymptotic variance, it is clear that p f 1 n pxq performs better than r f 1 n pxq and q f 1 n pxq. This is the reason why we have chosen to make use of p f 1 n pxq to estimate the derivative f 1 pxq for our real life data experiments.

We next focus our attention on simulations with sparse regions of the random design pX n q by choosing p " 0.8 in (2.20). One can observe in Figure 2.3 the sparse regions r0, 0.2s and r0.8, 1s. It was pointed out by [34] that local polynomial estimators with compactly supported kernels have shaky behaviour in sparse data areas. n pxq (dashed line) and p f 1LP n pxq (dotted line) with sparse region in r0, 0.2s and r0.8, 1s. The true curve of f 1 pxq is in solid line Finally, in order to underline the differences between p f 1 n pxq and p f 1LP n pxq, we compute the average MSE on three points : x " 0.12, x " 0.51 and x " 0.88 for N " 500 realizations.

The first and third points are located in sparse data areas, while the second one is in the dense center area. One can observe in Figure 2

High-frequency valvometry data

The motivation of this paper is to monitor sea shores water quality. For that purpose, we study bivalves activities by recording the valve movements. We use a high frequency, noninvasive valvometry electronic system developed by the UMR CNRS 5805 EPOC laboratory in Arcachon (France). The electronic principle of valvometry is described by [36], [7] and on the website http://molluscan-eye.epoc.u-bordeaux1.fr. This electronic system works autonomously without human intervention for a long period of time (at least one full year). Each animal is equipped with two light coils (sensors), of approximately 53mg each (unembedded), fixed on the edge of each valve. One of the coils emits a high-frequency, sinusoidal signal which is received by the other coil. The strength of the electric field produced between the two coils being proportional to the inverse of distance between the point of measurement and the center of the transmitting coil, the distance between coils can be measured and the accuracy of the measurements is a few µm.

For each sixteen animals, one measurement is received every 0.1s (10 Hz). So, the activity of oyster is measured every 1.6s and each day, we obtain 864 000 triplets of data : the time of the measurement, the distance between the two valves and the animal number.

A first electronic card in a waterproof case next to the animals manages the electrodes and a second electronic card handles the data acquisition. The valvometry system uses a GSM/GPRS modem and Linux operating system for the data storage, the internet access, and the data transmission. After each 24h period or any other programmed period of time, In the left hand side, relationship between the opening amplitude (in millimeters) and the time of the experiment (over 24 hours period). In the right hand side, the closing and opening velocity (millimeters per second) according to time (over the same period).

As argued in [2], [15] and [16], pollution can affect the activity of oysters and in particular the shells opening and closing velocities and so the movement speeds can be considered as an indicator of the animal stress activity since its movements are associated to aquatic system perturbations. In [2], the authors propose an interesting deterministic alternative method for the estimation of movement velocity based on differentiator estimators.

An example of valves activity and opening/closing velocity recordings June 2, 2011 is given in Figure 2.6. Moreover, as the recursive estimator (2.9) defined in random design regression have smaller asymptotic variance than for fixed design regression [4], we select the estimate p f 1 n of f 1 . Figure 2.7 displays for the same day the plot of the estimate p f 1 n of f 1 of the valve closing and opening velocity for one oyster at the Locmariaquer site. The bandwidth parameter 51 was selected by a standard the cross validation method. We propose in Figure 2.8 a visualization of the opening and closing velocity estimations of the 16 oysters for the period between the 63rd (March, 4th) and the 151st (May, 31th) of 2011. For each day, we compute for each oyster the estimator p f 1 n pxq of f 1 pxq for times x between 0 and 24h. Each velocity is represented by a color code : yellow for the smallest velocities, red for the highest and orange for the intermediate. Therefore, we can process the day velocity vector into a color line made of red, orange and yellow. We do it for each animal and superimpose multi-coloured dotted line. Hence, one day corresponds to 16 multi-coloured dotted lines. We repeat the process the day after and put the new 16 lines under the ones already obtained. This graphical representation reveals different clusters of global behaviors of the animals. First, one can notice yellow diagonal zones corresponding to the closed states of the animals. These states are highly correlated to the tidal amplitude, the animals being closed at low tide. Until the 100th day the animals have a normal behaviour but then we can observe a predominance of red in activity periods.

It can be explained by a sudden change in temperature in the environment associated to the modifications of the specific activity of two enzymatic biomarkers meaning a possible pollution as described in [16]. We have performed many other analyses of these data using extreme value theory and other nonparametric statistical methods, all of which point the same conclusion ( [9], [15], [16]). Altogether, we anticipate that this approach could have a significant contribution providing in situ instant diagnosis of the bivalves behavior and thus appears to be an effective, early warning tool in ecological risk assessment. The proofs of the almost sure convergence results rely on the following lemma. We also refer the reader to [32] for the estimation of the derivative of the Parzen-Rosenblatt estimator.

Lemma 2.5.1. Assume that pA 1 q, pA 2 q and pA 3 q hold. Then, the estimators p g n and p h n , given by (2.4), satisfy for any x P R,

lim
p h n pxq " 1 n n ÿ k"1 f pX k q h k K ˆx ´Xk h k ˙`1 n n ÿ k"1 ε k h k K ˆx ´Xk h k ˙.
Hence, by derivation, we have the decomposition

n p h 1 n pxq " A n pxq `Bn pxq (2.25)
where

A n pxq " n ÿ k"1 a k pxq " n ÿ k"1 f pX k qv k pX k , xq, B n pxq " n ÿ k"1 b k pxq " n ÿ k"1 ε k v k pX k , xq with v n pX n , xq " 1 h 2 n K 1 ˆx ´Xn h n ˙. (2.26)
On the one hand, we have for any x P R,

Era n pxqs " ż R f px n qv n px n , xqgpx n qdx n " 1 h n ż R f px ´hn yqgpx ´hn yqK 1 pyqdy. (2.27)
The regression function f as well as the density function g are bounded continuous and twice differentiable with bounded derivatives. Consequently, it follows from Taylor's formula that it exist θ f , θ g in the interval s0, 1r such that, for any x P R, f px ´hn yq " f pxq ´hn yf 1 pxq `h2 n y 2 2 f 2 px ´hn yθ f q, and gpx ´hn yq " gpxq ´hn yg 1 pxq `h2 n y 2 2 g 2 px ´hn yθ g q.

By a careful analysis of each term in the product f px ´hn yqgpx ´hn yq, we deduce from (2.27) together with assumption pA 1 q that Era n pxqs " ´`f pxqgpxq

˘1 ż R yK 1 pyqdy `hn f 1 pxqg 1 pxq ż R y 2 K 1 pyqdy `Rn pxq " `f pxqgpxq ˘1 `Rn pxq (2.28)
where the remainder R n pxq satisfies

sup xPR |R n pxq| " Oph n q.
Consequently, (2.28) immediately leads to

lim nÑ8 1 n ErA n pxqs " `f pxqgpxq ˘1, (2.29)
which is the limit we are looking for. By the same token,

Era 2 n pxqs " ż R f 2 px n qv 2 n px n , xqgpx n qdx n " 1 h 3 n ż R f 2 px ´hn yqgpx ´hn yq `K1 pyq ˘2dy " 1 h 3 n ξ 2 f 2 pxqgpxq `ζn pxq, (2.30)
where ξ 2 is defined in (2.15) and the remainder ζ n pxq is such that

sup xPR |ζ n pxq| " O ´1 h 2 n ¯.
Therefore, we deduce from (2.28) and (2.30) that

lim nÑ8 1 n 1`3α VarpA n pxqq " ξ 2 f 2 pxqgpxq 1 `3α . (2.31)
On the other hand, we assume that for any n ě 1, X n and ε n are conditionally independent given F n´1 where F n " σpX 1 , ε 1 , . . . , X n , ε n q. Consequently, we have for any x P R,

Erb n pxq|F n´1 s " Erε n v n pX n , xq|F n´1 s " Erε n |F n´1 sErv n pX n , xq|F n´1 s " Erε n |F n´1 sErv n pX n , xqs " 0.
Moreover,

Erb 2 n pxq|F n´1 s " Erε 2 n v 2 n pX n , xq|F n´1 s " Erε 2 n |F n´1 sErv 2 n pX n , xq|F n´1 s " σ 2 Erv 2 n pX n , xqs.
Furthermore, we have

Erv 2 n pX n , xqs " ż R v 2 n px n , xqgpx n qdx n " 1 h 3 n ż R gpx ´hn yq `K1 pyq ˘2dy " 1 h 3 n ż R
´gpxq ´hn yg 1 pxq `h2 n y 2 2 g 2 px ´hn yθ g q ¯`K 1 pyq ˘2dy

" 1 h 3 n ξ 2 gpxq `∆n pxq (2.32)
where ξ 2 is defined in (2.15) and the remainder ∆ n pxq is such that

sup xPR |∆ n pxq| " O ´1 h 2 n ¯.
Consequently, denoting Hence, we obtain from the strong law of large numbers for martingales given e.g. by Theorem 1.3.15 of [14] that, for any γ ą 0, pM A n pxqq 2 " opn 1`3α plog nq 1`γ q a.s. and pB n pxqq 2 " opn 1`3α plog nq 1`γ q a.s. Therefore, as 0 ă α ă 1{3, it ensures that, for any where

W n pxq " n ÿ k"1 v 2 k pX k ,
x P R lim nÑ8 1 n M A n pxq " 0 a.
C n pxq " n ÿ k"1 c k pxq " n ÿ k"1 f pX k q gpX k q v k pX k , xq, D n pxq " n ÿ k"1 d k pxq " n ÿ k"1 ε k gpX k q v k pX k , xq.
As in the proof of Lemma 2.5.1, we find that for any x P R such that gpxq ą 0,

lim nÑ8 1 n ErC n pxqs " f 1 pxq and lim nÑ8 1 n 1`3α VarpC n pxqq " ξ 2 f 2 pxq p1 `3αqgpxq . (2.38)
Hereafter, we split n q f 1 n pxq into three terms Appendix B. Proofs of the asymptotic normality results.

n q f 1 n pxq " M C n pxq `
In order to prove Theorem 2.3.2, we shall make use of the central limit theorem for martingales given by Theorem 2.1.9 of [14]. First of all, we focus our attention on convergence (2.18) since it is the easiest convergence to prove.

Proof of convergence (2.18). It follows from (2.39) that

a nh 3 n `q f 1 n pxq ´f 1 pxq ˘" a nh 3 n n `M C n pxq `ErC n pxqs `Dn pxq ´nf 1 pxq ˘,
which implies the martingale decomposition

a nh 3 n `q f 1 n pxq ´f 1 pxq ˘" 1 ? n 1`3α `xe, M n pxqy `q R n pxq ˘(2.42) where e " ˜1 1 ¸, M n pxq " ˜M C n pxq D n pxq ¸,
and the remainder q R n pxq " ErC n pxqs ´nf 1 pxq "

n ÿ k"1
`Erc k pxqs ´f 1 pxq ˘.

(2.43)

It follows from Taylor's formula that it exists θ f Ps0, 1r such that, for any x P R,

Erc n pxqs " ż R f px n qv n px n , xqdx n " 1 h n ż R f px ´hn yqK 1 pyqdy " f 1 pxq `hn 2 ż R f 2 px ´hn yθ f qy 2 K 1 pyqdy
sup xPR ˇˇq R n pxq ˇˇď τ 2 M f n ÿ k"1 h k .
However, it is easily seen that

n ÿ k"1 h k ď 1 1 ´α n 1´α .
Therefore, as soon as α ą 1{5, we obtain that sup xPR ˇˇq R n pxq ˇˇ" op ? n 1`3α q.

(2.45)

Hereafter, the predictable quadratic variation [14] of the two-dimensional real martingale pM n pxqq is given, for all n ě 1, by the diagonal matrix

xMpxqy n " ˜xM C pxqy n 0 0 xDpxqy n ¸.
Then, it follows from (2.40) that for any x P R such that gpxq ą 0,

lim nÑ8 1 n 1`3α xMpxqy n " ξ 2 p1 `3αqgpxq ˜f 2 pxq 0 0 σ 2 ¸. (2.46)
Furthermore, it is not hard to see that the martingale pM n pxqq satisfies the Lindeberg condition. As a matter of fact, we assume that the sequence pε n q has a finite conditional moment of order p ą 2. Let a ą 0 be such that p " 2p1 `aq. If we denote ∆M n pxq "

M n pxq ´Mn´1 pxq, we have for all n ě 1, We recall that p " 2p1 `aq. For any ε ą 0, if A k px, ε, nq " }∆M k pxq} ě ε ? n 1`3α ( , we have from (2.55),

E " }∆M n pxq} p |F n´1 ‰ " E "``∆ M C n pxq ˘2 ``∆D n pxq ˘2˘1 `a|F n´1 ‰ ď 2 a E "ˇˇ∆ M C n pxq ˇˇp `ˇ∆ D n pxq ˇˇp ˘|F n´1 ‰ . ( 2
1 n 1`3α n ÿ k"1 E " }∆M k pxq} 2 1 A k px,ε,nq |F k´1 ‰ ď 1 ε p´2 n b n ÿ k"1 E " }∆M k pxq} p |F k´1 ‰ ď M p ε p´2 n b n ÿ k"1 1 h 2p´1 k a.s. ď M p n c ε p´2 a.s.
where b " pa `1qp1 `3αq and c " apα ´1q. Since c ă 0, the Lindeberg condition is clearly satisfied. Finally, we can conclude from the central limit theorem for martingales [14] that for any x P R such that gpxq ą 0,

1 ? n 1`3α M n pxq L ÝÑN `0, Γpxq ˘, ( 2
p k pxq " n ÿ k"1 f pX k qu k pX k , xq, Q n pxq " n ÿ k"1 q k pxq " n ÿ k"1 ε k u k pX k , xq with
u n pX n , xq " 1 h n K ˆx ´Xn h n ˙.
Hence, for any x P R such that gpxq ą 0, we obtain from (2.10), (2.34) We saw in (2.28) that Era n pxqs " `f pxqgpxq ˘1 `Rn pxq where sup xPR ˇˇR n pxq ˇˇ" Oph n q. By the same token, ErP n pxqs " f pxqgpxq `ζn pxq where sup xPR ˇˇζ n pxq ˇˇ" Oph 2 n q. Therefore, as soon as α ą 1{5, we obtain that

sup xPR ˇˇr R n pxq ˇˇ" O ˜n ÿ k"1 h k ¸" op ? n 1`3α q.
(2.59) Furthermore, as in the proof of (2.35) and (2.46), the predictable quadratic variation of the four-dimensional real martingale pM n pxqq satisfies, for any x P R,

lim nÑ8 1 n 1`3α xMpxqy n " Γpxq (2.60)
where Γpxq is the four-dimensional covariance matrix given by Γpxq " ξ 2 gpxq p1 `3αq

¨f 2 pxq 0 0 0 0 σ 2 0 0 0 0 0 0 0 0 0 0 ‹ ‹ ‹ ‹ ' .
Moreover, via the same lines as in the proof of (2.55), we can also show that pM n pxqq satisfies the Lindeberg condition. Finally, we find from the central limit theorem for martingales [14] that for any x P R,

1 ? n 1`3α M n pxq L ÝÑN `0, Γpxq ˘,
which implies, from (2.58) and (2.59), that for any x P R such that gpxq ą 0,

a nh 3 n `r f 1 n pxq ´f 1 pxq ˘L ÝÑN ˆ0, ξ 2 1 `3α f 2 pxq `σ2 gpxq ˙.
Proof of convergence (2.16). First of all, for any x P R, denote hpxq " f pxqgpxq.

‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' , M n pxq " ¨MA n pxq B n pxq M V n pxq M U n pxq M P n pxq Q n pxq ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ '
, where the martingale difference sequences pM A n pxqq, pB n pxqq and pM P n pxqq, pQ n pxqq were previously defined in (2.34) and (2.57), while the martingale difference sequences pM U n pxqq and pM V n pxqq are given by M U n pxq " U n pxq ´ErU n pxqs and M V n pxq " V n pxq ´ErV n pxqs with

U n pxq " n ÿ k"1 u k pX k , xq " n ÿ k"1 1 h n K ˆx ´Xn h n ˙, V n pxq " n ÿ k"1 v k pX k , xq " n ÿ k"1 1 h 2 n K 1 ˆx ´Xn h n ˙.
It is not hard to see that the remainder p R n pxq, which can be explicitely calculated, plays a negligible role since, as soon as α ą 1{5, sup xPR ˇˇp R n pxq ˇˇ" op ? n 1`3α q.

(2.65)

It remains to establish the asymptotic behavior of the six-dimensional real martingale pM n pxqq. As in the proof of (2.46) and (2.60), we can show that for any x P R,

lim nÑ8 1 n 1`3α xMpxqy n " Γpxq (2.66)
where Γpxq is the six-dimensional covariance matrix given by Γpxq " ξ 2 gpxq p1 `3αq ¨f 2 pxq 0 f pxq 0 0 0 0 σ 2 0 0 0 0

f pxq 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' .
Moreover, via the same lines as in the proof of (2.55), pM n pxqq satisfies the Lindeberg condition. Hence, we obtain from the central limit theorem for martingales [14] that for

any x P R, 1 ? n 1`3α M n pxq L ÝÑN `0, Γpxq ˘. ( 2
‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' (2.68)
with ℓpxq " f pxqg 1 pxq ´f 1 pxqgpxq.

Finally, we deduce from (2.64), (2.65), (2.67) and (2.68) together with Slutsky's lemma that for any x P R such that gpxq ą 0,

a nh 3 n `p f 1 n pxq ´f 1 pxq ˘L ÝÑN `0, σ 2 pxq ˘(2.69)
where σ 2 pxq " xepxq, Γpxqepxqy. However, as hpxq " f pxqgpxq, it is not hard to see that

σ 2 pxq " ξ 2 p1 `3αqg 3 pxq ¨gpxq gpxq ´f pxqgpxq ‹ ‹ ' T ¨f 2 pxq 0 f pxq 0 σ 2 0 f pxq 0 1 ‹ ‹ ' ¨gpxq gpxq ´f pxqgpxq ‹ ‹ ' " ξ 2 p1 `3αqgpxq ¨1 1 ´f pxq ‹ ‹ ' T ¨f 2 pxq 0 f pxq 0 σ 2 0 f pxq 0 1 ‹ ‹ ' ¨1 1 ´f pxq ‹ ‹ ' " ξ 2 σ 2 p1 `3αqgpxq ,
which completes the proof of Theorem 2.3.2.

Chapitre 3

A nonparametric statistical procedure for the detection of marine pollution

Abstract. This paper is devoted to the estimation of the derivative of the regression function in fixed-design nonparametric regression. We establish the almost sure convergence as well as the asymptotic normality of our estimate.

We also provide concentration inequalities which are useful for small sample sizes. Numerical experiments on simulated data show that our nonparametric statistical procedure performs very well. We also illustrate our approach on high frequency environmental data for the study of marine pollution.

Key words. Environmental statistics and data analysis ; mathematical statistics ; nonparametric estimation ; fixed-design regression

Introduction

Marine ecosystems are facing all over the world the impact of multiple stressors due to seashore pollution and climatic change. Water quality should be dealt as a major concern in our modern society. Marine pollution comes mostly from land based sources and this pollution can lead to the collapse of coastal ecosystems and have public health impacts.

In this context, there is an urgent need to elaborate real-time reliable sensors in order to supervise the water quality within a decision making process. Among these sensors, bio-indicators are increasingly used and are highly effective to reveal pollutions or perturbations in the aquatic systems. For example bivalve mollusks, such as oysters, mussels, and giant clams, are relevant sentinel organisms to evaluate water quality. These bioindicators can also be considered as biological responses to climate change influencing water-quality and biological organisms. One of the main challenges is to determine how bio-indicators are affected by pollution and climate change.

The high frequency measurements of valve activity in bivalves using valvometry is a possible way to study their behaviors as well as a global analysis of possible perturbations due to the environment. The idea of valvometry system is to use the bivalves ability to close its shell when exposed to a contaminant as an alarm signal [10]. Therefore, the continuous monitoring of the gap between the two valves is an efficient way to study their behavior when facing water pollution [13,27]. This system measures at high frequency the bivalve shell movements. Nowadays, valvometric techniques allow autonomously longterm recordings of valve movements without interfering their normal behavior and produce massive volume of data.

Several statistical models have been developed to analyze this huge amount of data [3,7,11,12]. The main objective is to propose optimized statistical procedures in order to detect the world's oceans changes. Here, we propose a statistical procedure to estimate the opening and closing velocities of bio-indicators. As a matter of fact, in an inhospitable environment, bio-indicators behavior is altered. Consequently, detecting changes of the closing and opening velocity can provide insights about the health of bio-indicators and so can be used as bio-indicators of the water quality.

Nonparametric regression processes were provided to modelize the opening/closing activity. The goal was to estimate the regression function governing the relationship between time and amplitude. We refer the reader to [9,14,33] for some excellent books on nonparametric kernel estimation of densities and regression functions. Here, we shall focus our attention on the estimation of the derivative of the regression function using a recursive version of the Nadaraya-Watson estimator [1,18,25,26,36]. In a recent work, [4] investigated the asymptotic properties of this estimate in a random-design regression framework.

The recursive estimate is really powerful as soon as data come one by one. Indeed, each new value provides an additional information that can sharpen the estimation of the regression function. It is not necessary to compute again all the values using all the data.

However, in most cases in valvometry, data are collected in one batch. We don't have access to the information until midnight where we obtain the last 24-hour measurements.

The recursive nature of the estimation is then not necessary. Consequently, we propose an alternative strategy based on a fixed-design regression framework with a more efficient and easy to handle statistical procedure.

Another strategy for the recursive estimation of the regression function and of its derivatives is to make use of the local polynomial fitting approach [21,34,35]. The advantages of local polynomial fitting are twofold. The estimates are easily computable and they have nice asymptotic properties. However, we will see in the sequel that the main drawback of this approach is that it requires the kernels to be compactly supported. This is really important as the asymptotic variance of our estimates can be ten times smaller for the Gaussian kernel than for all other kernels with compact supports. Furthermore, a natural question is the extension of our results in a more general setting of dependent data. In all the previous literature on the estimation of the regression function for short-range dependent, long-range dependent or strong mixing data [2,17,22,18], it is necessary to assume that the kernels are compactly supported. It seems really difficult to get rid of this assumption. This is the reason why we have chosen to restrict ourselves to the case of independent data and to favor our statistical application on the detection of marine pollution.

The paper is organized as follows. Section 3.2 describes the estimation of the first derivative of the regression function. It also describes the data sets compatible with the methodology developed in this paper. We establish in Section 3.3 the pointwise almost sure convergence, the asymptotic normality as well as concentration inequalities of our estimate. Section 3.4 is devoted to numerical experiments in order to study the performance of our nonparametric statistical procedure. Section 3.5 presents an application for the survey of marine water using high-frequency valvometry. All the technical proofs are postponed to Sections 3.6, 3.7 and 3.8.

Estimation of the velocity

In order to record the bivalve activity, each animal is equipped with two electrodes.

The relationship between the distances of the two electrodes pY n q and the times of the measurement pX n q is given by the nonparametric regression satisfying, for all n ě 1,

Y n " f pX n q `εn (3.1)
where pε n q is the random error sequence and pX n q is a sequence of independent and identically distributed random variables. Our purpose is to estimate the derivative of the unknown regression function f which is directly associated with the velocity of the valve opening/closing activities of the animals. In [4], a statistical procedure based on the derivative of the recursive Nadaraya-Watson estimator was implemented. The regression function f was estimated, for any x P R, by

p f n pxq " n ř k"1 Y k h k K ˆx ´Xk h k ṅ ř k"1 1 h k K ˆx ´Xk h k ˙, (3.2)
where the kernel K is a chosen probability density function and the bandwidth ph n q is a sequence of positive real numbers decreasing to zero.

As previously argued, the data are collected in one batch. Then, it is not necessary to make use of a recursive estimator of f . In addition to that, when we observe a group of 16 animals, one measurement is received every 0.1 seconds. So, the activity of one animal is measured every 1.6 seconds. Hence, the times of measurement pX n q are perfectly known.

Consequently, we are in a fixed design case since the differences between X n and X n`1 are always the same. For all k " 1, . . . , n, denote t k " k{n where n is the number of measurement taken over the whole day. Therefore, the model described in (3.1) can be rewritten, for all k " 1, . . . , n, as

Y k " f pt k q `εk . (3.3)
The nonparametric regression function f is estimated, for any x Ps0, 1r, by

p f n pxq " 1 nh n n ÿ k"1 Y k K ˆx ´tk h n ˙. (3.4)
Since we are interested in the velocity, we will investigate the asymptotic behavior of the derivative of this estimator given, for any x Ps0, 1r, by

p f 1 n pxq " 1 nh 2 n n ÿ k"1 Y k K 1 ˆx ´tk h n ˙.
(3.5)

Theoretical results

In order to investigate the asymptotic behavior of p f 1 n pxq, it is necessary to introduce several classical assumptions.

pA 1 q The kernel K is either the Gaussian kernel or a positive symmetric bounded function compactly supported, twice differentiable with bounded derivatives,

such that ż R Kpxqdx " 1, ż R K 1 pxqdx " 0, ż R xK 1 pxqdx " ´1.
pA 2 q The regression function f is bounded continuous, twice differentiable with bounded derivatives.

pA 3 q The noise pε n q is a sequence of independent and identically distributed random variables with zero mean and finite positive variance σ 2 .

Furthermore, the bandwidth ph n q is a sequence of positive real numbers, decreasing to zero, such that nh n tends to infinity. For the sake of simplicity, we shall make use of

h n " 1{n α with 0 ă α ă 1.
Our first result on the almost sure convergence of our estimate is as follows.

Theorem 3.3.1. Assume that pA 1 q, pA 2 q and pA 3 q hold. Then, for any x in s0, 1r, as soon as α ă 1{3, we have the pointwise almost sure convergence

lim nÑ8 p f 1 n pxq " f 1 pxq a.s. (3.6)
Démonstration. The proof is given in Appendix 2.5.

Remark. A similar result was previously established by Gasser and Müller [15] for compactly supported kernels. However, our approach is different from the one of Gasser and

Müller which relies on Hoeffding inequality. One can observe that in Theorem 2 of [15],

it is necessary to assume that 2α ă 1 ´1{p where pε n q has a finite moment of order p ě 2. Hence, for p " 2, the almost sure convergence given in [15] only holds in the more restrictive case α ă 1{4. Our approach relies on the law of iterated logarithm for weighted sums of independent random variables [16] and only finite variance is required for pε n q.

Our second theoretical result is devoted to the asymptotic normality for p f 1 n pxq. Denote

ξ 2 " ż R `K1 pxq ˘2dx.
Theorem 3.3.2. Assume that pA 1 q, pA 2 q and pA 3 q hold. Then, for any x in s0, 1r, we have as n tends to infinity the pointwise asymptotic normality

a nh 3 n `p f 1 n pxq ´E" p f 1 n pxq ‰˘L ÝÑN `0, σ 2 ξ 2 ˘. (3.7)
Furthermore, as soon as 1{5 ă α ă 1{3 we also have as n tends to infinity for any x in s0, 1r,

a nh 3 n `p f 1 n pxq ´f 1 pxq ˘L ÝÑN `0, σ 2 ξ 2 ˘. (3.8)
Démonstration. The proof is given in Appendix 2.5.

Remark. The asymptotic normality was previously investigated by Gasser and Müller [15] for compactly supported kernels. We shall see in the next Section that the asymptotic variance ξ 2 is ten times smaller for the Gaussian kernel than for all other kernels with compact supports. On the same vein, let r f n pxq be the recursive local polynomial estimator of f pxq proposed by Vilar and Vilar [34] in the random-design framework. It was proven in [34] that

a nh 3 n `r f 1 n pxq ´f 1 pxq ˘L ÝÑN ˆ0, p1 ´2αq 2 ζ 2 p1 ´αqgpxq σ 2 ẇhere ζ 2 " ˆżR x 2 Kpxqdx ˙´2 ż R x 2 `Kpxq ˘2dx.
However, it was assumed in [34] that the kernel K is compactly supported. For example, for the Epanechnikov kernel, one can easily check that ζ 2 " 15{7 » 2.1429. However, it is easy to see that for the standard Gaussian kernel, ξ 2 " 1{p4 ? πq » 0.1410. Consequently, our approach offers much more flexibility in the choice of the kernel K, and the asymptotic variance for p f 1 n pxq is smaller than that of r f 1 n pxq. We are also brought to the same conclusion in the case of dependent data [2,17,22,18].

Our last result deals with concentration inequalities for our estimate, in the spirit of McDiarmid's inequality [23]. We wish to stress that this result is true for small or large sample sizes as it holds whatever the value of n is. Denote

Λ " sup xPR ˇˇK 1 pxq ˇˇand ζ " ż R ˇˇK 1 pxq ˇˇdx.
Theorem 3.3.3. Assume that pA 1 q, pA 2 q and pA 3 q hold. Moreover, assume that one can find a positive constant M such that, for all 1 ď k ď n, | Y k | ď M a.s. Then, for any x in s0, 1r and for any positive t,

P ´ˇp f 1 n pxq ´E" p f 1 n pxq ‰ˇˇě t ¯ď 2 exp ´´nh 2 n t 2 2M 2 Λ 2 ¯, (3.9)
and

P ´ˇˇż R ˇˇp f 1 n pxq ´f 1 pxq ˇˇdx ´E" ż R ˇˇp f 1 n pxq ´f 1 pxq ˇˇdx ıˇˇˇě t ¯ď 2 exp ´´nh 2 n t 2 2M 2 ζ 2 ¯. (3.10)
Démonstration. The proof is given in Appendix 3.8.

Simulation results

This section is devoted to numerical experiments in order to evaluate the performances of our statistical estimation procedure and to assess more specifically the sensitivity of our approach when the assumptions of Theorems 3.3.1 and 3.3.2 are not satisfied. We shall also consider heavy-tailed distributions for the driven noise pε n q such as the wellknown Pareto distribution. We have also simulated the short range dependance situation observed in Section 3.5 via an autoregressive process of order 1.

The data are generated by the nonparametric regression given, for all k " 1, . . . , n, by

Y k " f pt k q `εk , (3.11)
where t k " k{n, the regression function f is defined, for all x in r0, 1s, by f pxq " px `2q sinp4πx 2 q `2 sinp8πxq (3.12) and the noise pε n q is sequence of independent identically distributed random variables.

We implement our statistical procedure with sample size n " 10 000 since we have large datasets in the application described in Sections 3.5 and 3. Kpxq "

1 ? 2π exp ´´x 2 2 ¯. (3.13)
The derivative f 1 pxq is given, for all x in r0, 1s, by f 1 pxq " sinp4πx 2 q `8πxpx `2q cosp4πx 2 q `16π cosp8πxq.

It is well-known that the choice of the bandwidth h n " 1{n α is crucial in nonparametric kernel estimation. In order to select an automatic choice of α, we use a standard cross validation method which leads to α " 0.3034. After selecting α by cross validation, the left hand side of Figure 3.2 shows that the estimator p f 1 n pxq approaches very well the true derivative f 1 pxq for the Np0, 1q noise distribution, as well as for the Laplace Lp0, 1q and Pareto Pp3q noise distributions, in agreement with assumptions of Theorem 3.6. On the other hand, for the Pareto Pp3{2q noise distribution, the estimate p f 1 n pxq is quite far from the true derivative f 1 pxq. Furthermore, one can observe boundary effects for x close to 0 and 1. A wide literature is available on how to remove boundary effects, see e.g. the data reflection method given in [24]. In order to illustrate the pointwise asymptotic normality of our estimate, we implement a simulation study based on 2 000 realizations. We numerically check, for the Np0, 1q and the Pareto Pp3q noise distributions, the asymptotic normality at point x " 0. in the asymptotic variance of Z n pxq, we do the same simulation still with σ 2 " 1 but this time using different kernels. We choose the Epanechnikov, Cosine, Quartic and Triweight kernels, respectively given by Kpxq "

3 4 `1 ´x2 ˘1|x|ď1 , Kpxq " π 4 cos ´π 2 x ¯1|x|ď1 , Kpxq " 15 16 `1 ´x2 ˘21 |x|ď1 , Kpxq " 35 32 `1 ´x2 ˘31 |x|ď1.
The corresponding values of ξ 2 are respectively 1.5, 1.522, 2.14 and 3.18 . In Figure 3.4, one can see that the behavior of the variance of Z n pxq is the same for all values of x in s0, 1r and clearly depends on ξ 2 . One can also observe that the smaller asymptotic variance is obtained with the Gaussian kernel. To assess more specifically the sensitivity of our approach in the presence of autocorrelated errors, we have simulated the short range dependance observed in Section 3.5 using an autoregressive process of order 1. The data are still generated from the nonparametric regression model given by (3.11) with the same regression function given by (3.12). Howe-ver, the random noise pε n q is autocorrelated. We have chosen a stationary autoregressive process of order 1, given by

ε n " ρ ε n´1 `ξn
where the parameter |ρ| ă 1 and the innovations pξ n q are independent random variables sharing the same N p0, 1q distribution. We have for any k ě 0, corrpε n , ε n`k q " ρ k . Consequently, the autocorrelation in the errors goes down geometrically as the distance between them goes up. We observed in Figure 3.5 that even for this autocorrelated noise, our simulations yields satisfactory results for the pointwise almost sure convergence to the true derivative f 1 pxq. n pxq with ρ " 0.135 (dashed), ρ " 0.368 (dotted) and ρ " 0.607 (dot-dashed). The solid line is true derivative f 1 pxq.

Finally, our simulations yield satisfactory results of our estimation even in the presence of short range dependence and heavy-tailed distribution.

High-frequency valvometry data

Our goal is now to propose a way to survey the marine water quality. The use of biological indicators is of great interest for the detection of seashore pollution. These bio-indicators can also be considered as biological responses to climate change influencing water-quality and biological organisms. The main challenge is to determine how bio-indicators are affected by pollution and climate change. For that purpose, we study bivalves activities using a high-frequency noninvasive valvometry electronic system developed by the EPOC team in Arcachon, France, see [6] and [32] as well as the website http://molluscan-eye.epoc.u-bordeaux1.fr.

This electronic system records valve movements autonomously for a long period of time (at least one full year). Each animal is equipped with two light coils (sensors), of approximately 53mg each (unembedded), fixed on the edge of each valve. One of the coils emits a high-frequency, sinusoidal signal which is received by the other coil. The strength of the electric field produced between the two coils being proportional to the inverse of the distance between the point of measurement and the center of the transmitting coil, the distance between coils can be measured and the accuracy of the measurements is a few µm.

For each sixteen animals, one measurement is received every 0.1s (10 Hz). So, the activity of animals is measured every 1.6s and each day, we obtain 864 000 triplets of data : the times of the measurement, the distances between the two valves and the animal numbers.

A first electronic card in a waterproof case next to the animals manages the electrodes and a second electronic card handles the data acquisition. The valvometry system uses a GSM/GPRS modem and a Linux operating system for the data storage, the internet access, and the data transmission. After each 24-hour period or any other programmed period of time, the data are transmitted to a workstation server and then inserted into a SQL database which is accessible with the software R or a text terminal.

Several valvometric systems have been installed around the world : southern lagoon of New Caledonia (South Pacific), Spain, Ny Alesund Svalbard at 1300 km from the north pole, the north east of Murmansk in Russia on the Barents sea and at several sites in France with various species. We focus our attention on the IORO reef in the Havannah channel (GPS coordinates 166 ˝57 1 25 2 E, 22 ˝23 1 15 2 S) in the southern lagoon of New Caledonia.

The New Caledonia climatic regime is characterized by two main periods, an austral rainy summer (December-March) and a drier austral winter (May-October), each separated by transition phases. The water temperature closely follows air temperature changes. We study the opening/closing velocity of sixteen giant clams Hippopus hippopus placed in a single bag.

As argued in [11,12], environmental perturbations such as a pollution can affect the activity of biosensors and in particular the shells opening and closing velocities. For instance, we observed that a stressed animals due to the presence of pollution or environmental perturbations exhibits irregular and numerous micro-closing and opening periods with changes in the velocities in comparison with the normal situation. Consequently, the movement velocities can be considered as an indicator of the animal reaction activity since its movements are associated with aquatic system perturbations. Different statistical methods have been developed to analyse these high-frequency data [19,30,28,29,3,7] among others and a nonparametric adaptive estimator was proposed for extreme tail probabilities and quantiles [11,12]. In [3], a four-state stochastic process was considered to give inferences about oysters' health and to provide some arguments about the healthiness of their environment. The authors especially delineate links between groups of oysters and features related to the survey of the environment in different experimental sites, such as environmental variations. These methods also exhibit a link between the tide and oysters' behavior, as shown in [30]. In [7], a data-driven bandwidth choice for a kernel density estimator is investigated assuming the number of modes to be known and this methodology is illustrated using valvometry data. When considering these data, important intermittent activity at high frequency, with frequent and sudden "microclosing" events (meaning partial closures), at apparently random times and with random amplitudes is observed [29]. A fractal shot noise modeling is then proposed for quantifying and characterizing the behaviors of bioindicators directly exposed to their natural environment and exposed to changes in the water quality of their natural environment. Using valvometry data, a nonparametric quantile regression model was used to model in situ with accuracy giant clam growth rate behavior in relation to temperature [28] showing that the shell growth was significantly correlated to rising sea surface temperature. As the measurements were performed at 10 Hz for one among the sixteen animals, the developed platform for valvometry are able to measure the position of the opening of a mollusk shell. So, this system allows the bivalves to be studied in their natural environment with minimal experimental constraints with accuracy and the velocity measurements are enough accurate for our ve-locity estimations. An example of valves activity and opening/closing velocity recordings is depicted in Figure 3.6 for the 8th of July 2008. Because the data representation in Figures 3.6 and 3.7 are "noisy", we do not use the finite difference method to determine the velocity. We use instead the smooth nonparametric estimation of the derivative of the regression function described in Section 3.2.

We do not consider the fastest opening and closing velocities because the measurements every 1.6s do not allow us to estimate them with accuracy. Indeed, if an animal quickly closes or opens its shells, the 1.6 second gap between two measurements is too important to determine the velocity. To improve the higher velocities estimations, it is possible for a new study to use higher frequency data acquisition (for instance at 100 Hz). Here, we focus our attention on smaller velocities. Figure 3.7 shows the very good fit of fixed-design regression derivative p f 1 n pxq to the observed velocities. We have chosen to restrict ourselves to opening/closing velocities smaller than 0.3 millimeters per second as higher velocities can be seen as statistically insignificant outliers. In order to study the behavior of these bio-indicators with respect to the sea water temperature, we compare for the 16 bivalves the closing and opening velocities on two periods corresponding to the warmest period (from 10th of March to 2th of April 2008) and to the coldest period (July 2008). We focus here our attention on the opening velocities since the same trends are observed for the closing velocities.

There is some type of autocorrelation in the residuals confirmed by Box-Pierce, Ljung- The present work researches the effect of environmental conditions on giant clams in New Caledonia, focusing on a particular species that we believe to be more amenable than others to an online analysis of behavioral activity. However, the function a n is Lipschitz in the second variable. More precisely, for any fixed x in r0, 1s, there exists some positive constant L such that, for all y and z in r0, 1s, On the other hand, we also have

ˇˇa
ż xℓn px´1qℓn K 1 ptqdt " K `x ℓ n ˘´K `px ´1qℓ n ˘.
If K is a kernel with compact support, then for n large enough, this integral vanishes.

In addition, if K is the Gaussian kernel, K `x ℓ n ˘" oph n q and K `px ´1qℓ n ˘" oph n q. It ensures that restrictive assumption α ă 1{4 if we use of the law of the single logarithm for weighted sums of independent random variables given by Li and Tomkins [20], [31]. Finally, we deduce from the decomposition (3.14) together with (3.17) and (3.28) that

ż xℓn px´1qℓn K 1 ptqdt " oph n q. ( 3
lim nÑ8 p f 1 n pxq " f 1 pxq a.s.
which completes the proof of Theorem 3.3.1.

Proof of the asymptotic normality

We are now in the position to establish the pointwise asymptotic normality (3.7

Proof of the concentration inequalities

We shall follow the same approach as [8] who was the first to make use of McDiarmid's inequality [23] in nonparametric density estimation. For any x in r0, 1s, denote

p f 1 n pxq " p f 1 n px, Y 1 , . . . , Y n q " 1 nh 2 n n ÿ k"1 Y k K 1 ˆx ´tk h n ˙.
addition, assume that |Z k | ď M a.s. We clearly have

ˇˇp f 1 n px, Y 1 , . . . , Y n q ´p f 1 n px, Z 1 , . . . , Z n q ˇˇ" 1 nh 2 n ˇˇY k ´Zk ˇˇˇˇˇˇK 1 ˆx ´tk h n ˙ˇˇˇ, ď 2M Λ nh 2 n a.s.
Consequently, (3.9) immediately follows from McDiarmid's inequality [23], see also [9] or [5]. By the same token,

ˇˇˇż R ˇˇp f 1 n px, Y 1 , . . . , Y n q ´f 1 pxq ˇˇdx ´żR ˇˇp f 1 n px, Z 1 , . . . , Z n q ´f 1 pxq ˇˇdx ˇˇď ż R ˇˇp f 1 n px, Y 1 , . . . , Y n q ´p f 1 n px, Z 1 , . . . , Z n q ˇˇdx, ď 1 nh 2 n ż R ˇˇY k ´Zk ˇˇˇˇˇˇK 1 ˆx ´tk h n ˙ˇˇˇd x, ď 2M ζ nh 2 n a.s.
Finally, we obtain (3.10) once again from McDiarmid's inequality [23], which completes the proof of Theorem 3.3.3.

Introduction

Nowadays, the need to study huge amount of data implies the development of faster computation methods. In the case of regression estimation, the use of recursive estimator fulfil this demand when the observations are gathered sequentially (see [5], [19], [10] [4], [13]).

Indeed, with a non recursive estimator the arrival of a new data needs to recalculate the estimate from the whole data while a recursive estimator only need the previous value of the estimate.

In some cases, it is necessary to investigate the derivative of these estimators. In [1] and [2],

a statistical procedure was implemented to estimate the velocity of valve opening/closing activities of the shells of bivalve molluscs (oysters, mussels, giant clams) based on the derivative of the regression function estimation. In particular, [1] investigated the asymptotic behaviour of the derivative of the recursive Nadaraya-Watson estimator ( [9], [14]).

In this paper, we propose a different strategy which relies on the Révész estimator of a regression function . This estimator was introduced by Révész [12] and studied in [11], [6], [7], [8]. The aim of this paper is to study the asymptotic behaviour of the derivative of the Révész estimator and to compare its asymptotic variance to the asymptotic variance of the derivative of the recursive Nadaraya-Watson estimator.

The paper is organized as follows. In Section 4.1, we present our nonparametric estimation procedure of the derivative of the regression function. We establish in Section 4.2 we establish the pointwise almost sure convergence as well as the asymptotic normality of our estimate and compare the latter to the derivative of the recursive Nadaraya-Watson estimator. Section 4.3 deals with simulated data and the proofs of the theoretical results can be found in Section 4.4.

Let pX 1 , Y 1 q, ¨¨¨, pX n , Y n q be n couples of random variables related by the nonparametric regression given, for all n ě 1 by

Y n " f pX n q `εn (4.1)
where ε n are unknown random errors. In all the sequel, we assume that pX n q is a sequence of independent and identically distributed random variables with positive probability density function g. Our purpose is to estimate the derivative of the unknown regression function f using the derivative of the Révész estimator defined for all x P R, by setting f 0 pxq P R and, for n ě 1, by

fn pxq " fn´1 pxq `1 nh n K ˆx ´Xn h n ˙´Y n ´f n´1 pxq ¯, (4.2)
where the kernel K is a probability density function and the bandwidth h n is a sequence of positive real numbers decreasing to zero. For the sake of simplicity, we denote for all

x P R and n ě 1,

K n pxq " 1 h n K ˆx ´Xn h n ˙and K 1 n pxq " 1 h 2 n K 1 ˆx ´Xn h n ˙.
Hence, (4.2) can be rewritten as

fn pxq " ˆ1 ´1 n K n pxq ˙f n´1 pxq `1 n K n pxqY n
and its derivative is given, for any x P R, by

p f 1 n pxq " ˆ1 ´1 n K n pxq ˙f 1 n´1 pxq `1 n K 1 n pxq ´Yn ´f n´1 pxq ¯. (4.3)

Theoretical results

In order to investigate the asymptotic behavior of p f 1 n pxq, it is necessary to introduce several classical assumptions. pA 1 q The kernel K is a positive symmetric, bounded function, differentiable with bounded derivative, satisfying ż

R Kpxqdx " 1, ż R K 1 pxqdx " 0, ż R xK 1 pxqdx " ´1, ż R x 2 K 1 pxqdx " 0. ż R x 2 Kpxqdx " τ, ż R x 3 K 1 pxqdx " ζ, ż R K 1 pxq 2 dx " ξ 2 .
pA 2 q The regression function f and the density function g are bounded, continuous, three times differentiable with bounded derivatives.

pA 3 q The noise pε n q is a sequence of independent and identically distributed random variables with zero mean and finite positive variance σ 2 .

Furtermore, the sequences pX n q and pε n q are independent.

pA 4 q The bandwidth ph n q is a sequence of positive real numbers, decreasing to zero, such that nh n tends to infinity. For the sake of simplicity, we make use of h n " n ´α with 0 ă α ă 1{3.

pA 5 q For all x P R,

max ˆgpxq 2α , 2gpxq 1 ´3α ˙ą 1.
We denote, for all x P R,

Zpxq " Z 1 pxq `Z2 pxq with Z 1 pxq " f pxqζ K 1 pxq `f 1 pxqζ K pxq ´ζfK 1 pxq where ζ K pxq " 1 2 g 2 pxqτ, ζ K 1 pxq " 1 6 g p3q pxqζ, ζ f K 1 pxq " 1 6
`f pxqg p3q pxq `gpxqf p3q pxq `3f 1 pxqg 2 pxq `3g 1 pxqf 2 pxq ˘ζ and Z 2 pxq " g 1 pxq gpxqf 2 pxq `2f 1 pxqg 1 pxq 2pgpxq ´2αq τ.

We first give the bias and the variance of p f 1 n pxq.

Theorem 4.2.1. Assume that pA 1 q ´pA 5 q hold. For all x P R such that gpxq ą 0,

E " p f 1 n pxq ı ´f 1 pxq " $ ' ' ' ' & ' ' ' ' % h 2 n ˆZpxq gpxq ´2α `op1q ˙if gpxq ą 2α o ˜1 a nh 3 n ¸if gpxq ď 2α (4.4) and Var ´p f 1 n pxq ¯" $ ' & ' % 1 nh 3 n ˆgpxqσ 2 ξ 2 2gpxq ´p1 ´3αq `op1q ˙if 2gpxq ą 1 ´3α o ph 4 n q if 2gpxq ď 1 ´3α. (4.5)
As noticed in [7], one can observe that if 2gpxq ď 1 ´3α, the variance is negligible in front of the square of the bias. However, if gpxq ď 2α, the square of the bias become negligible with respect to the variance. In the case where gpxq ą 2α and 2gpxq ą 1 ´3α

we have to observe the value of the window parameter α. If α ă 1{7, then the variance is negligible in front of the squared bias while if 1{7 ď α ă 1{3, the squared bias is asymptotically dominated by the variance. Now, let us focus on the asymptotic normality of our estimator.

Theorem 4.2.2. Assume that pA 1 q ´pA 5 q hold. For all x P R such that gpxq ą 0,

• if α ă 1{7, then, as soon as gpxq ą 2α, One can notice that in the case where the pX n q have a standard uniform distribution, resulting in g " 1, the asymptotic variance for the derivative of the Révész estimator is the same as the asymptotic variance for the derivative of the recursive Nadaraya-Watson estimator [1]. However, if g ‰ 1, the asymptotic variance of the derivative of the Révész estimator is smaller as soon as x is such that gpxq P rp1 ´3αq{p1 `3αq, 1s.

Simulated data

This section is devoted to numerical experiments in order to evaluate the performances of our derivative estimate. First, we will observe the asymptotic behaviour of our estimator in order to illustrate (4.7) and (4.

Y n " f pX n q `εn , (4.9)
where the regression function f and its derivative f 1 are defined, for all x in r0, 1s, by f pxq " sinp2πx 3 q 3 and f 1 pxq " 18π x 2 cosp2πx 3 q sinp2πx 3 q 2 . (4.10)

The source of variation pε n q is a sequence of independent and identically random variables sharing the same Np0, 1q distribution. In order to illustrate Remark 4.2.1 we choose different distribution types for the random observation pX n q

• Up0, 1q,

• Ep3q : the number of data decrease as x increase,

• Beta p2, 2q : more data in the center,

• Beta p0.5, 0.5q : more data on the edges.

In [1], we implemented our statistical procedure with a large sample size since we had large datasets in the application. To make a proper comparison between the two estimators we use the same scale and choose n " 15 000 as sample size. One can see on Figure 4.1 an illustration of n couples pX, Y q following the model (4.9). We first illustrate the pointwise almost sure convergence of the estimator p f 1 n to f 1 . We make use of a Gaussian kernel since it minimizes the integral ξ 2 defined in (4.8) and therefore the asymptotic variance. In order to select the parameter α of the bandwidth we use the cross validation method which leads to the choice of α " 0.32 n pxq (dotted line) to f 1 pxq (solid line) with pX n q following an Ur0, 1s distribution (dashed line) and an Ep3q distribution (dotted line) on the left and pX n q following Betapα, βq distributions on the right with α " β " 2 (dashed line) and α " β " 0.5 (dotted line). the recursive Nadaraya-Watson estimator. Let us denote

σ 2 R " gpxqξ 2 σ 2 2gpxq ´p1 ´3αq , σ 2 N W " ξ 2 σ 2 p1 `3αqgpxq and r σ " σ 2 R σ 2 N W
where σ 2 R is the asymptotic variance found in (4.8) and σ 2 N W is the asymptotic variance of the derivative of the recursive Nadaraya-Watson estimator [1]. To study the behavior of r σ , we choose ten points between 0 and 1 and for each point x we generate an pX n , Y n q sample from which we compute both the derivative of the Révész estimator and the derivative of the recursive Nadaraya-Watson estimator. We realize this operation N " 200 times and we compute the variances of a nh 3 n ´p f 1 n pxq ´f 1 pxq ¯for the two estimators in order to obtain a ratio rσ . Figure 4.3 shows that when pX n q is uniformly distributed over the interval r0, 1s we obtain, for a similar choice of α, the same asymptotic variances. However, for other cases, the ratio obtained by simulations approaches the true value of r σ and illustrate Remark 4.2.1. Therefore, as soon as x is such that gpxq ą 1, the choice of the derivative of the Révész estimator is a better choice in term of asymptotic variance.

Proofs

Before giving the outlines of the proofs, we start with the following straightforward calculations which will be useful in all the sequel. Under pA 1 q and pA 2 q, we have E rf pX n qK 1 n pxqs "

1 h 2 n ż R f px n qK 1 ˆx ´xn h n ˙gpx n q dx n , " 1 h n ż R f px ´yh n qgpx ´yh n qK 1 pyq dy.
Using Taylor's formula, we have for any x P R, f px ´yh n q " f pxq ´hn yf Zpxq " Z 1 pxq `Z2 pxq.

E rf pX n qK 1 n pxqs " f pxqgpxq h n ż R K 1 pyqdy ´pf 1 pxqgpxq `f pxqg 1 pxqq ż R yK 1 pyqdy `hn ˆf pxqg 2 pxq 2 `f 2 pxqgpxq 2 `f 1 pxqg 1 pxq ˙żR y 2 K 1 pyqdy ´ζfK 1 pxqh 2 n `O `h3 n ˘. Since ż R K 1 pyqdy " ż R y 2 K 1 pyqdy " 0 this expectation reduces to E rf pX n qK 1 n pxqs " f 1 pxqgpxq `f pxqg 1 pxq ´ζfK 1 pxqh 2 n `o `h2 n ˘. ( 4
Then, using Lemma 1 of [7], we find that

D n pxq " Zpxq gpxq ´2α h 2 n `o `h2 n ˘.
Moreover, if gpxq ď 2α, we have, from assumption pA 5 q that 4α ą 1 ´3α. Hence, it is not hard to see that h Therefore, D n pxq " D n´1 pxq p1 ´dn `opd n qq `dn opw n q where w n " a γ n h ´3 n .

Hence,using once again Lemma 1 of [7] we obtain that

D n pxq " o ´aγ n h ´3 n ¯.
According to the variance decomposition formula we have,

V n pxq " Varp p f 1 n pxqq " V

Perspectives

D'un point de vue statistique, l'une des suites à mener pour ces travaux pourrait être le calcul de la convergence presque sûre de la dérivée de l'estimateur de Révész. Cette étude impliquerait une étude du comportement asymptotique de l'estimateur de Révész qui intervient dans la dérivée.

  Je tiens tout d'abord à remercier mes directeurs pour leur encadrement tout au long de cette thèse. Merci à Bernard pour m'avoir fait confiance et poussé à me lancer dans cette aventure depuis le Master, pour avoir su être exigeant tout en me laissant libre d'explorer toutes les pistes que je voulais. Merci à Gilles qui malgré la distance puis la très longue distance aura toujours su m'aiguiller, pour son aide et son accueil au cours de mes séjours à Vannes. Je tiens à remercier Fabienne Comte et Gérard Biau pour avoir pris le temps de lire en détail mon manuscrit ainsi qu'Aurore Delaigle pour avoir accepté d'être membre du jury et à Jérémie Bigot pour avoir accepté de le présider. Merci à Jean-Charles Massabuau, non seulement pour sa présence dans le jury mais aussi pour son initiation à la valvométrie, grâce à laquelle j'ai pu m'immerger dans cette thèse avec encore plus de motivation. Je remercie par la même occasion Mohamedou Sow pour son accueil et ses explications techniques. Je tiens aussi à remercier Marc Arnaudon pour m'avoir mis le pied à l'étrier au cours d'un stage très enrichissant et m'avoir appris que la recherche était un processus long, ce conseil aura été d'un grand secours de nombreuse fois au cours de cette thèse. Merci aussi à toutes les personnes qui m'auront aidé pour les enseignements : Vincent, Jérémie, Michel, Adrien et Frédérique en particulier. Je remercie le personnel administratif de l'IMB et du LMBA, notamment Karine et Rachel pour les cours d'oenologie et Véronique Vellet pour avoir facilité mes séjours en Bretagne. J'en profite aussi pour remercier les membres du LMBA pour leur accueil chaleureux.

  Merci aux Béarnais et Béarnaises, Pauline pour ce nouvel an reconstructeur, Patrick, Elizabeth, Jade et Ambre pour ces vacances toujours agréables. Merci à Shehrazad qui, grâce à cette thèse, revient sur les lieux de notre rencontre après sept ans d'une magnifique amitié. Enfin, je tiens à remercier plus particulièrement Flora, Mélissa et Radia pour leurs encouragements et leur soutien autour d'aligot, de bubble-tea ou pendant un mariage kabyle. Je remercie tous les Bordelais et Bordelaises. A commencer par Charlotte pour m'avoir fait découvrir la ville. Sans toi j'aurais sûrement commis l'énorme erreur d'opter pour Toulouse. Merci à tous les doctorants de l'IMB qui auront égayé toutes ces longues journées pluvieuses entre pauses cafés et feuilletés au fromage du Haut-Carré. Les anciens d'abord, Zoé la plus cool de IMB, Samuel qui partage mon amour de Sergio Ramos et Pierre ma haine de la Juve. Merci surtout à Bruno pour m'avoir intégré à coup de verre de vin rouge et de soirée Champions League (Inch'Allah Metz croupit en Domino's Ligue 2 jusqu'à la fin des temps). Merci aux Italiens : Sergio, Roberto, Marco, Corentin (Italien par cobureauité) et Edo pour toutes ces soirées au Cock et grâce à qui le moindre Sassuolo-SPAL pouvait devenir un sujet bouillant. Je ne sais pas si je retiendrais tous les résultats de ma thèse mais je me souviendrai toujours qu'il n'y a que Milan qui n'a pas battu Benevento en 2017/2018. Merci à Paul pour ses conseils pronostics en tennis et Krisztian pour avoir patiemment supporté ces discussions. Merci à Mohammed (et Zaza), Louis-Marie et Luis pour les parties de ping-pong au Labri, à Manon et Guillaume (au moment où j'écris ça Payet vient de mettre une reprise de volée monstrueuse, c'est un signe).

1 Introduction 1 . 1

 111 Valvométrie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Enjeux environnementaux . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Bioindicateurs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Bivalves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Valvométrie HFNI . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Estimateurs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Estimation non paramétrique d'une fonction de densité . . . . . . . 1.2.2 Estimation non paramétrique d'une fonction de régression . . . . . 1.2.3 Dérivée des estimateurs . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Résultats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Article 1 : Nonparametric recursive estimation of the derivative of the regression function with application to sea shores water quality 1.3.2 Article 2 : A nonparametric statistical procedure for the detection of marine pollution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Article 3 : On the asymptotic behavior of the derivative of the Révész estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Nonparametric recursive estimation of the derivative of the regression function with application to sea shores water quality 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nonparametric estimation of the derivative . . . . . . . . . . . . . . . . . . 2.3 Theoretical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 High-frequency valvometry data . . . . . . . . . . . . . . . . . . . . . . . .

3 A

 3 nonparametric statistical procedure for the detection of marine pollution 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1

 11 représente l'anatomie d'une huître, l'un des bivalves les plus communs, disponible sur le site du Comité National de Conchyliculture. D'autres représentant des bivalves sont les moules, les bénitiers ou les coquilles Saint-Jacques. Ces animaux font partie de la classe des coquillages filtreurs. Pour se nourrir ou pour respirer, elle aspire l'eau de mer puis la filtre grâce à ses branchies pour capter les particules nécessaires à son alimentation et l'oxygène nécessaire à sa respiration. Ils peuvent filtrer ainsi près de 15 litres d'eau par heure.

Figure 1 . 1 -

 11 Figure 1.1 -Anatomie d'une huître. Source : CNC

Figure 1 . 2 -

 12 Figure 1.2 -Schéma d'un système de mesure de valvométrie mesurant l'ouverture d'unepalourde en fixant une valve. Source :[9]

Figure 1 . 3 -Figure 1 . 4 -

 1314 Figure 1.3 -Schéma d'un système de mesure de valvométrie utilisant l'induction éléctromagnétique. 1-Moule ; 2-Electrodes ; 3-support ; 4-socle. Source : [17]

Figure 1 . 5 -

 15 Figure 1.5 -Carte des différents sites de valvométrie HFNI sur le littoral atlantique

Figure 1 . 6 -

 16 Figure 1.6 -Carte des différents sites de valvométrie HFNI dans le monde

Figure 1 . 8 -Figure 1 . 9 -Figure 1 . 10 -

 1819110 Figure 1.8 -Représentation du comportement d'une huître sur le site de Locmariaquer (Morbihan) au cours d'une journée. remarque deux périodes distinctes : une période fermée et une période ouverte. Cette période ouverte est rythmée par de nombreux mouvements des valves. Ces battements permettent à l'animal de créer un courant d'eau pour respirer et se nourrir. La Figure 1.9 représente le signal propre à une ponte. Il se caractérise par une succession de mouvements rapides pour propulser les oeufs. Surveiller la durée des pontes, leur répétitions au cours des années est fondamental pour estimer du bon ou mauvais état de santé d'une population. Il est aussi possible de dater la mort d'un animal en notant la date de dernière activité du muscle adducteur qui se traduit par la fin dernière fermeture sur les enregistrement, comme indiqué dans la Figure 1.10. Si la mort d'un animal peut ne pas être préoccupante, la mort d'un groupe dans une période courte et dont les perturbations du comportement ont toutes démarrées à la même période peut signaler une perturbation. Toutes ces informations ainsi que de nombreuses données valvométriques sont accessibles sur le site

Figure 1 . 12 -Figure 1 . 13 -

 112113 Figure1.12 -Normalité asymptotique de p f 1 n pxq (première ligne), r f 1 n pxq (deuxième ligne) aux points x " 0.5 (colonne de gauche) et au point x " 0.8 (colonne de droite). Les courbes de densité représentent les distributions asymptotiques théoriques

1 .Figure 1 . 14 -Figure 1 . 15 -

 1114115 Figure 1.14 -Représentations des variances asymptotiques pour différents types de noyaux. De haut en bas : Cubique, Quadratique, Epanechnikov et Gaussien. Les lignes horizontales correspondent aux vraies valeur de ξ 2

Figure 2 .Figure 2 . 1 -

 221 Figure2.1 shows that the three estimators p f 1 n pxq, r f 1 n pxq and q f 1 n pxq approximate well the true derivative f 1 pxq after selecting α " 0.3 by cross validation.

Figure 2 . 2 -Figure 2 . 3 -Figure 2 . 4 -

 222324 Figure 2.2 -Asymptotic normality of p f 1 n pxq (first row), r f 1 n pxq (second row) and q f 1 n pxq (third row) at point x " 0.5 (left column) and point x " 0.8 (right column). The density curves represent the asymptotic normal distributions given in Theorem 2.3.2.

Figure 2 . 6 -

 26 Figure 2.6 -A typical example of valvometric data for one oyster the June 2, 2011. In the left hand side, relationship between the opening amplitude (in millimeters)and the time of the experiment (over 24 hours period). In the right hand side, the closing and opening velocity (millimeters per second) according to time (over the same period).

Figure 2 . 7 -

 27 Figure 2.7 -The dashed line displays for June 2, 2011, the estimated f 1 pxq using estimator p f 1 n pxq versus the time x and the solid lines represent the observed speeds of valve openings and closings. The closing and opening velocity are measured in millimeters per second.

Figure 2 . 8 -

 28 Figure 2.8 -Velocities estimation using p f 1 n pxq from the 63th to the 151th days of 2011, considering the 16 oysters in Locmariaquer. The x-axis represents the time in a 24 hour time period and the y-axis represents the number of days since January 1, 2011.

3 . 2 .Figure 3 . 1 -

 3231 Figure3.1 -Simulated data pt k , Y k q for k " 1, . . . , n and n " 10 000. The noise pε k q has a Np0, 1q distribution on the top left, a Laplace Lp0, 1q distribution on the top right and a Pareto Ppθq distribution with shape parameter θ " 3 on the bottom left and θ " 3{2 on the bottom right. The solid line is the true function f given by (3.12).

Figure 3 . 2 -

 32 Figure 3.2 -Illustration of the pointwise almost sure convergence of p f 1 n pxq for the Np0, 1q (dashed line), the Laplace Lp0, 1q (dot-dashed line) and the Pareto Pp3q (dotted line) noise distributions on the left and the Pareto Pp3{2q (dotted line) noise distribution on the right. The solid lines represent the true derivative f 1 pxq.

2 DensityFigure 3 . 3 -

 233 Figure 3.3 -Illustration of the asymptotic normality at point x " 0.2 for the Np0, 1q, the Pareto Pp3q, and the Pareto Pp3{2q noise distributions. The density curve represents the asymptotic Np0, σ 2 ξ 2 q distribution.

Figure 3 . 4 -

 34 Figure 3.4 -Representation of the asymptotic variances of Z n pxq with different kernels. From top to bottom : Triweight, Quartic, Cosine, Epanechnikov and Gaussian. The horizontal dashed lines correspond to the values of ξ 2 .

Figure 3 . 5 -

 35 Figure 3.5 -Illustration of the almost sure convergence of p f 1n pxq with ρ " 0.135 (dashed), ρ " 0.368 (dotted) and ρ " 0.607 (dot-dashed). The solid line is true derivative f 1 pxq.

Figure 3 . 6 -

 36 Figure 3.6 -A typical example of valvometric data for one giant clam the 8th of July 2008. On the left-hand side, relationship between the opening amplitude and the time of the experiment. On the right-hand side, the closing and opening velocity with respect to time.

Figure 3 . 8 -

 38 Figure 3.8 -ACF of the residuals of the model (3.1).

Figure 3 . 9 -

 39 Figure 3.9 -Histograms of the derivative estimators p f 1 n pxq : On the left-hand side in red for the warmest period and on the right-hand side in blue for the coldest period in New Caledonia.

Figure 3 .

 3 Figure 3.10 -Quantile-Quantile plot of the derivative estimator p f 1 n pxq during the two selected periods : in abscissa the coldest period and in ordinate the warmest period in New Caledonia.

  8). Then we will compare the performance of the derivative of the Révész estimator with the performance of the derivative of the recursive estimator of Nadaraya-Watson as argued in Remark 4.2.1. The data are generated by the nonparametric regression given by

Figure 4 .2 shows that the estimator p f 1 nFigure 4 . 1 -Figure 4 . 2 -

 414142 Figure 4.1 -Illustration of the pX n , Y n q, following the model (4.9), where the observation pX k q has an Ur0, 1s distribution on the top left, an Ep3q distribution on the top right and a Betapα, βq distribution with parameter α " β " 2 on the bottom left and α " β " 0.5 on the bottom right. The true curve of f is in plain line.

Figure 4 . 3 -

 43 Figure 4.3 -Illustration of the behavior of rσ . The observation pX n q has an Ur0, 1s distribution on the top left, an Ep3q distribution on the top right and a Betapα, βq distribution with parameter α " β " 0.5 on the bottom left and α " β " 2 on the bottom right. The true value of r σ in each case is in dashed line.

2 n

 2 " o `aγ n h ´3 n ˘which leads to R D 1,n pxq " o ´aγ n h ´3 n ¯and R D 2,n pxq " o ´aγ n h ´3 n ¯.

  Au cours de cette thèse nous avons comparé les performances asymptotiques de cinq estimateurs non paramétriques de la dérivée d'une fonction de régression.Il a été montré dans le premier article que parmi les trois estimateurs issus de l'estimateur de Nadaraya-Watson, le plus performant en termes de variance asymptotique était la dérivée de la version récursive de l'estimateur de Nadaraya-Watson. Pourtant cet estimateur nécessite d'estimer la densité g des données alors que pour les deux autres estimateurs (Johnston et Wand-Jones) la densité g est connue. L'apport d'information sur g augmenterait donc la variance asymptotique.Cette conclusion apparaît aussi dans le deuxième article où nous utilisions un modèle plus adapté aux données de valvométrie. En prenant en compte le caractère déterministe des données de temps, nous obtenons une variance asymptotique plus grande que celle de la dérivée de la version récursive de l'estimateur de Nadaraya-Watson.Dans le troisième article nous montrons que la dérivée de l'estimateur de Révész a une plus petite variance asymptotique que la version récursive de l'estimateur de Nadaraya-Watson lorsque x est tel que gpxq P rp1´3αq{p1`3αq, 1s. Dans le cadre de l'application en valvométrie nous sommes dans un cas où g " 1 et donc où les variances asymptotiques sont identiques. L'obtention de la convergence presque sûre pour la dérivée de l'estimateur de Nadaraya-Watson récursif ainsi que sa plus grande rapidité en terme de temps de calculs nous pousserait à préférer son utilisation.Un résumé de ces résultats peut être trouvé dans la Figure(5.1).

Figure 5 . 1 -

 51 Figure 5.1 -Comparaison des comportements asymptotiques des différents estimateurs.

  . . Pour effectuer un contrôle sanitaire sur les eaux de baignade par exemple, des échantillons d'eau sont prélevés toutes les deux semaines puis analysés en laboratoire. On ne dispose donc d'aucune informations sur la qualité de l'eau entre deux prélèvements et augmenter le nombre de mesures augmenterait les coûts de ces contrôles sanitaires. De plus, les analyses effectuées concernent uniquement la concentration de germes et de parasites. Tester d'autres paramètres à chaque étude se heurterait à une nouvelle limite de budget.

Une méthode permettant de contourner ces contraintes est l'utilisation de bioindicateurs.

Un bioindicateur est une espèce animale ou végétale dont la surveillance des caractéristique fournit une indication sur la qualité écologique de son milieu. Ainsi, l'observation du comportement d'un animal permet une surveillance globale et continue de son environnement. Un des exemples les plus anciens est l'utilisation des canaris dans les mines pour détecter la présence de gaz toxiques. Étant plus sensibles que les hommes, leur évanouissement ou leur mort donnait un signal d'alarme sur la qualité de l'air. Aujourd'hui,

  Dans cette thèse, nous considérons que l'ouverture Y n entre deux valves et le temps X n où est pris la mesure sont reliés par le modèle de régression non paramétrique donné pour tout n ě 1, parY n " f pX n q `εn (1.1) où ε n est le terme d'erreur et f la fonction de lien inconnue. Ce modèle a été utilisé avec succès au cours de différents travaux d'étude du mouvement des valves ([29]. Ici, comme nous étudions la vitesse, nous nous concentrerons sur l'estimation de la dérivée de la fonction de lien f . Dans cette partie, nous présenterons les différents estimateurs

	1.2 Estimateurs

étudiés au cours de cette thèse. Nous commencerons par un bref historique des différents estimateurs non paramétrique d'une fonction de densité et de régression.

  .5 that while both estimators share the same MSE in the central region, the recursive Nadaraya-Watson estimator has smaller MSE in the sparse data areas for all kernels. A more elaborate comparison is desirable, and it should be performed someday.

	Design Epanechnikov kernel Quartic kernel	Gaussian kernel
	points	RNW	RLP	RNW	RLP	RNW	RLP
	x=0.12	0.1056	0.4378	0.1801 0.4951	0.3891 0.5865
	x=0.51	0.3945	0.4429	0.6695 0.5215	0.7119 0.7031
	x=0.88	0.0429	0.2942	0.0675 0.3228	0.1429 0.3871
	Figure 2.5 -MSE associated to the Recursive Nadaraya-Watson (RNW) and Recur-
	sive Local Polynomial (RLP) estimators for design points x " 0.12 (sparse region), x " 0.51 (dense region) and x " 0.88 (sparse region) with Epa-nechnikov, Quartic and Gaussian kernels.

  Démonstration. We shall only prove the almost sure convergence (2.24) inasmuch as (2.21) and (2.22) are well-known and the proof of (2.23) is more easy to handle and follow the same lines as the proof of (2.24). We deduce from (3.1) and (2.4) that for any x P R,

	lim nÑ8	p h n pxq " f pxqgpxq	a.s.	(2.22)
	Moreover, as soon as 0 ă α ă 1{3, we also have for any x P R,
	lim nÑ8	p g 1 n pxq " g 1 pxq	a.s.	(2.23)
	lim nÑ8	p h 1 n pxq " `f pxqgpxq ˘1	a.s.	(2.24)
	nÑ8	p g n pxq " gpxq	a.s.	(2.21)

  A n pxq ´ErA n pxqs. One can observe that pM A n pxqq and pB n pxqq are both square integrable martingale difference sequences with predictable quadratic variations respectively given by xM A pxqy n " VarpA n pxqq and xBpxqy n " σ 2 ErW n pxqs. Consequently,

	(2.25) can be rewritten as
		n p h 1 n pxq " M A n pxq `ErA n pxqs `Bn pxq,	(2.34)
	where M A n pxq " (2.31) together with (2.33) immediately lead to
	lim nÑ8	xM A pxqy n n 1`3α "	ξ 2 f 2 pxqgpxq 1 `3α	and	lim nÑ8	xBpxqy n n 1`3α "	σ 2 ξ 2 gpxq 1 `3α	.	(2.35)
								xq,
	it follows from (2.32) that
			lim nÑ8	1 n 1`3α ErW n pxqs "	ξ 2 gpxq 1 `3α	.	(2.33)
	We are now in the position to prove the almost sure convergence (2.24). The decomposition

  Proof of Theorem 2.3.1. We shall now proceed to the proof of the Theorem 2.3.1. It clearly follows from relation (2.9) and Lemma 2.5.1 that for any x P R such that gpxq ą 0,

	It only remains to prove (2.14). We obtain from relation (2.11) that
				n q f 1 n pxq " C n pxq `Dn pxq	(2.37)
			lim nÑ8	p h 1 n pxq " `f pxqgpxq ˘1	a.s.
	Thus Lemma 2.5.1 is proven.
	lim nÑ`8	p f 1 n pxq " lim nÑ`8 " f 1 pxqgpxq `f pxqg 1 pxq ´f pxqg 1 pxq ´p h 1 n pxq p g n pxq ´p h n pxqp g 1 n pxq p g 2 n pxq ¯" `f pxqgpxq ˘1 gpxq gpxq " f 1 pxq	´f pxqgpxqg 1 pxq g 2 pxq a.s.	a.s.
	By the same token, relation (2.10) and Lemma 2.5.1 immediately lead to
		lim nÑ`8	r f 1 n pxq " lim nÑ`8	´p h 1 n pxq gpxq	´p h n pxqg 1 pxq gpxq 2 ¯" f 1 pxq	a.s.

s. and lim nÑ8 1 n B n pxq " 0 a.s. (2.36) Finally, we deduce from decomposition (2.34) together with (2.29) and (2.36) that for any x P R,

  One can observe that pM C n pxqq and pD n pxqq are both square integrable martingale difference sequences with predictable quadratic variations satisfying, for any x P R such that gpxq ą 0,

	lim nÑ8	xM C pxqy n n 1`3α "	ξ 2 f 2 pxq p1 `3αqgpxq	and	lim nÑ8	xDpxqy n n 1`3α "	ξ 2 σ 2 p1 `3αqgpxq	.	(2.40)
	Therefore, we deduce from the strong law of large numbers for martingales that, as soon
	as 0 ă α ă 1{3,
		lim nÑ8	1 n	M C n pxq " 0 a.s.	and		lim nÑ8	1 n	D n pxq " 0 a.s.	(2.41)
	Finally, it follows from (2.39) together with (2.38) and (2.41) that for any x P R such that gpxq ą 0, lim nÑ8 q f 1 n pxq " f 1 pxq a.s.
	which achieves the proof of Theorem 2.3.1.

ErC n pxqs `Dn pxq (2.39) where M C n pxq " C n pxq ´ErC n pxqs.

  , where v n is defined in (2.26). Since f 2 is bounded, we have

	Hence, we deduce from (2.43) and (2.44) that
			sup xPR	ˇˇErc n pxqs ´f 1 pxq ˇˇď M f τ 2 h n	(2.44)
	where	M f " sup xPR	|f 2 pxq|	and	τ 2 "	2 1	ż

R

y 2 ˇˇK 1 pyq ˇˇdy.

  n px, yq ´an px, zq ˇˇď As a matter of fact, f is a differentiable function on r0, 1s with bounded derivative, which implies that f is Lipschitz. By the same token, K 1 is also Lipschitz. Since f and K 1 are two bounded functions, the product a n is Lipschitz in the second variable. Moreover, if K is a kernel with compact support, K 1 is also compactly supported. It means that, for any x ‰ y in r0, 1s, and for n large enough, a n px, yq vanishes. Hence, the Lipschitz constant for a n does not depend on n. Furthermore, if K is the Gaussian kernel, it is well-known 1{3, the two sequences are equivalent. It remains to carefully investigate the asymptotic behavior of B n pxq. The regression function f is bounded continuous and twice differentiable with bounded derivatives. Consequently, it follows from Taylor's formula with integral remainder that, for all x in r0, 1s and for On the one hand, as the bandwidth h n goes to zero, ℓ n tends to infinity. Hence, for any x in s0, 1r, xℓ n goes to `8, while px ´1qℓ n goes to ´8. Consequently, we obtain that

	which implies that
	L h n x´hnt where n goes to infinity as soon as α ă any t P R, ˇˇ1 h 2 n A n pxq ´1 h 2 n B n pxq ˇˇď L 2nh 3 n Hence, as nh 3 f px ´hn tq " f pxq ´hn tf 1 pxq `ż x `s ´x `hn t ˘f 2 psqds. ż 1 B n pxq " 0 a n px, yqdy. |y ´z|.	(3.19) (3.21)
	that Therefore, B n pxq can be rewritten as sup tPR ˇˇK 1 ptq ˇˇ" 1 ? 2πe Consequently, for any all y and z in r0, 1s, B n pxq " ż 1 0 f pyqK 1 ˆx ´y h n ˙dy " h n and ż xℓn px´1qℓn sup tPR f px ´hn tqK 1 ptqdt, ˇˇK 2 ptq ˇˇ" 1 ? 2π . ż xℓn ż xℓn ˇˇˇK 1 ˆx ´y h n ˙´K 1 ˆx ´z h n ˙ˇˇˇď 1 ? 2π |y ´z| h n " h n f pxq px´1qℓn K 1 ptqdt ´h2 n f 1 pxq px´1qℓn tK 1 ptqdt `Rn pxq	(3.22)
	where ℓ n " 1{h n and the remainder R n pxq is given by leading to ˇˇa n px, yq ´an px, zq ˇˇď L f |y ´z| ? 2πe `Mf ? 2π where L f is the Lipschitz constant for f and R n pxq " h n ż xℓn px´1qℓn ż x x´hnt `s ´x `hn t ˘f 2 psqK 1 ptqdsdt. |y ´z| h n	(3.20)
			M f " sup tPR	|f ptq|.
		ż xℓn	ż
	Therefore, (3.19) immediately follows from (3.20). Hereafter, we obtain from (3.18) and (3.19) that, for all x in r0, 1s, lim nÑ8 px´1qℓn tK 1 ptqdt " R tK 1 ptqdt " ´1. (3.23)
	ˇˇA n pxq	´ż 1 0	a n px, yqdy ˇˇď "	L h n L h n	n ÿ k"1 n ÿ k"1	ż t k t k´1 ż t k t k´1	|t k ´y|dy, pt k ´yqdy,
			"	L 2h n	n ÿ k"1	pt k ´tk´1 q 2 ,

  Since pε n q is a sequence of independent random variables with mean zero and variance px, t k qq 2 " o `VarpC n pxqq ˘.(3.34)The kernel K is bounded, twice differentiable with bounded derivatives. Hence, there exists some positive constant M K such that, for any x in s0, 1r,If K is a kernel with compact support, then for n large enough, Q n pxq vanishes. In addition, if K is the Gaussian kernel, it follows from standard Gaussian calculation that

	). It follows (3.33) Consequently, in order to apply Lindeberg-Feller's central limit theorem to the sequence σ 2 , we clearly have VarpC n pxqq " σ 2 n ÿ k"1 pc n px, t k qq 2 " σ 2 nh n Σ n pxq. Hence, we obtain from (3.30) that lim nÑ8 VarpC n pxqq " σ 2 ξ 2 . pC n pxqq, it is only necessary to check that max 1ďkďn where ∆ n pxq " 1 h 2 n ˘. `An pxq ´Bn pxq However, it follows from (3.21) that there exists some positive constant L such that ˇˇ∆ n pxq ˇˇď ż `8 Kptqdt " 1 xℓ n ȃnd Kpxℓ n q `1 `op1q xℓn L . 2nh 3 n Hence, as soon as, 0 ă α ă 1{3 lim nÑ8 a nh 3 n ∆ n pxq " 0. ż px´1qℓn ´8 Kptqdt " Kppx ´1qℓ n q `1 `op1q ˘. ´1 px ´1qℓ n It implies that for any x in r0, 1s, (3.37) lim nÑ8 a nh 3 n Q n pxq " 0. (3.40) Furthermore, we deduce from the decomposition of B n pxq given in (3.22) that 1 h 2 n B n pxq ´f 1 pxq " 1 h n f pxqP n pxq ´f 1 pxqQ n pxq `1 h 2 n R n pxq (3.38) where Moreover, we immediately infer from (3.25) that, as soon as α ą 1{5 lim nÑ8 a nh 3 n 1 h 2 n R n pxq " 0. (3.41) pc n max 1ďkďn pc n px, t k qq 2 ď M 2 K nh n . P n pxq " ż xℓn px´1qℓn K 1 ptqdt and Q n pxq " 1 `ż xℓn px´1qℓn Finally, it follows from (3.38) together with (3.39), (3.40) and (3.41) that tK 1 ptqdt. lim nÑ8 a nh 3 n ´1 h 2 n B n pxq ´f 1 pxq ¯" 0 On the one hand, It clearly implies (3.34) as nh n goes to infinity as soon as n does. Therefore, (3.32) together with (3.33) leads to a nh 3 n `p f 1 n pxq ´Er p f 1 which is exactly convergence (3.7). It now remains to prove (3.8) replacing Er p f 1 f 1 pxq in (3.35). We clearly have for all x in s0, 1r a nh 3 n `p f 1 n pxq ´f 1 pxq ˘" a nh 3 n `p f 1 n pxq ´Er p f 1 On the other hand, an integration by parts for Q n pxq leads to n pxqs ˘`D n pxq have for any x in r0, 1s, lim nÑ8 nh 3 n 1 h n P n pxq " 0. (3.39) a n pxqs by Hence, if K is a kernel with compact support or if K is the Gaussian kernel, we clearly n pxqs ˘L ÝÑN `0, σ 2 ξ 2 ˘(3.35) a nh 3 n 1 h n P n pxq " a nh n `Kpxℓ n q ´Kppx ´1qℓ n q ˘. which ensures that D n pxq goes to zero as n tends to infinity, completing the proof of Theorem 3.3.2.
	from (3.3) and (3.5) that a nh 3 n ´p f 1 n pxq ´Er p f 1 n pxqs ¯" C n pxq and, for any x and y in s0, 1r, c n px, yq stands for where c n px, yq " 1 ? nh n K 1 ˆx where D n pxq " a nh 3 n `Er p f 1 n pxqs ´f 1 pxq C n pxq " ´y h n ˙. ˘. Consequently, in order to prove (3.8), we only have to show that D n pxq goes to zero as n n ÿ k"1 c n px, t k qε k (3.32) tends to infinity. We already saw in Appendix 2.5 that Er p f 1 n pxqs ´f 1 pxq " 1 h 2 n A n pxq ´f 1 pxq " 1 h 2 n B n pxq ´f 1 pxq `∆n pxq (3.36) Q n pxq " 1 ´ż xℓn px´1qℓn ı xℓn Kptqdt `"tKptq , px´1qℓn " ż R Kptqdt ´ż xℓn px´1qℓn ı xℓn Kptqdt `"tKptq , px´1qℓn " ż `8 xℓn Kptqdt `ż px´1qℓn ´8 ı xℓn Kptqdt `"tKptq px´1qℓn .

  First of all, we focus our attention on the evaluation of the bias(4.4). Denoting E rK 1 n pxqf pX n qs ´E rK 1 n pxqs f pxq ´E rK n pxqs f 1 pxq.

	In the same vein, we also obtain Since pX n q is a sequence of independent random variables, we also have Var pK n pxqq " o ˆ1 h n ˙(4.16) R D " ı 2,n pxq " E rK 1 n pxqs E f pxq ´p f n´1 pxq .
	and It was observed in [7] that Moreover, for the sake of simplicity, we will denote CovpK n pxq, K 1 n pxqq " o γ n " E " fn pxq ı ´f pxq " $ & % ζ E pxqh 2 n `o ph 2 ˆ1 h n n q if gpxq ą 2α ˙. o `aγ n h ´1 n ˘if gpxq ď 2α 1 n . where ζ E pxq " gpxqf 2 pxq `2f 1 pxqg 1 pxq τ. 2pgpxq ´2αq If gpxq ą 2α, we obtain that	(4.17)
	4.4.1 Proof of Theorem 3.1 R D 2,n pxq " Z 2 pxqh 2 n		`o `h2
		D n pxq " E	" f 1 p n pxq ı	n ´f 1 pxq, `o `h2 n ˘.	(4.13)
	In addition to that, we obtain with similar computations, that E " pf pX n qK 1 n pxqq 2 ı " 1 h 3 n `f pxq 2 gpxqξ 2 `op1q which, we can easily obtain from (4.3) that D n pxq " D n´1 pxq p1 ´γn ErK n pxqsq `γn `RD 1,n pxq `RD Zpxq gpxq h 2 n 2,n pxq ˘(4.18) with
	where	in view of (4.11), implies that
	Var pf pX n qK 1 n pxqq " R D 1,n pxq " E rK 1 n pxq pY n ´f pxqqs ´E rK n pxqs f 1 pxq 1 h 3 n `f pxq 2 gpxqξ 2 `op1q ˘. E " pK 1 n pxqq 2 ı " 1 h 3 n which, (4.14) R D 2,n pxq " E " K 1 n pxq ´f pxq ´p f n´1 pxq ¯ı . `gpxqξ 2 `op1q in view of (4.13), implies that Moreover and We deduce from pA 3 q that
	Var pK 1 n pxqq " 1,n pxq " Hence, it follows from (4.11) together with (4.12), and (4.13) that 1 h 3 n `gpxqξ 2 `op1q ˘. R D	(4.15)
		R D 1,n pxq " Z 1 pxqh 2 n	`o `h2 n ˘.	(4.19)

.11)

Following the same lines, we also find that E rK n pxqs " gpxq ´ζK pxqh 2 n `o `h2 n ˘(4.12) and E rK 1 n pxqs " g 1 pxq ´ζK 1 h 2 n which, combined to (4.18) and (4.19), leads to D n pxq " D n´1 pxq p1 ´dn `opd n qq `dn u n p1 `op1qq where d n " γ n gpxq and u n "

  Then, it follows from (4.14) together with (4.15) and (4.16) that Var ´p f 1 n pxq| p f 1 n´1 pxq, p f n´1 pxq ¯" γ 2 n h ´3 n ˆgpxqξ 2 ˆσ2 `´p f n´1 pxq ´f pxq Consequently, with (4.12), we find thatV 2 pxq " p1 ´2γ n gpxq `opγ n qq Var ´p f 1 n´1 pxq ¯. O ph 2 n q if 2gpxq ą 1 ´α, o pnh 7 n q if 2gpxq ď 1 ´α.Therefore, since we are in the case where α ą 1{7, we obtain that n pxq p1 ´αn pxqq Cov pR n´1 pxq, ρ n pxqq " 0 and therefore p1 ´αn pxqq Cov pR n´1 pxq, ρ n pxqq " o `γn h ´3 n ˘. Varpρ n pxqq " o `γn h ´3 n ˘. (4.43) In view of (4.40), and since we obtain from (4.29) that p1 ´αn pxqq 2 " p1 ´2γ n gpxq `opγ n qq , (4.44) we have, using (4.42) and (4.43), that V R n pxq " p1 ´2γ n gpxq `opγ n qq V R n´1 pxq `γn o `γn h ´3 n ˘. Furthermore, we obtain from (4.33) that H n pxq " p1 ´αn pxqq H n´1 pxq `γn η n pxq. Hence, denoting V H n pxq " Var pH n pxqq, we have V H n pxq " p1 ´αn pxqq 2 V H n´1 pxq `γ2 n Varpη n pxqq. (4.47) It remains to study the asymptotic behavior of Varpη n pxqq. It follows from (4.28) that Varpη n pxqq " Var pK n pxqq E Since we are in the case where 2gpxq ą 1 ´3α and α ą 1{7, we obtain from Theorem Consequently, (4.47) together with (4.44) leads to V H n pxq " p1 ´2γ n gpxq `opγ n qq V H n´1 pxq `γn o `γn h ´3 nHence, the study of the asymptotic behavior of a nh 3 n ∆ n pxq can be reduced to the study of the asymptotic behaviour of a nh 3 n A ´1 n pxqM n pxq. Furthermore, using (4.38), the law of large number for square integrable martingales ensures that, as soon as the Lindeberg condition is met, ) follows from the combination of (4.50) and (4.51). It remains to prove that the martingale pM n pxqq satisfies the Lindeberg condition, that is for all ℓ ą 0, |∆M k |2 1 t|∆M k |ěℓ ? an |F k´1 ‰ P ÝÑ0 (4.52) where a n " γ n h ´3 n A 2 n pxq and ∆M n " ∆M n pxq " γ n A n pxqµ n pxq. First, one can easily see that for any d ą 0 " p1 ´p2p1 `dqγ n `o pγ n qq Λ n´1 pxq `γn E n pxq Hence, the key point is the study of E " |µ n | 2p1`dq |F n´1 ‰ . In view of (4.26), we have µ n pxq " K 1 n pxq `pf pX n q ´f pxqq `εn `´f pxq ´p f n´1 pxq ¯˘´f 1 pxqK n pxq `Rµ n pxq. (4.56) Since d ą 0,there exists a constant C d such that | 2p1`dq |K 1 pyq| 2p1`dq gpx ´yh n q dy Finally, we can easily observe, using (4.11) together with (4.13) and (4.12), that R µ n pxq " g 1 pxq ´f pxq ´p f n´1 pxq ¯˘`O `h2 ˘1`d Λ n pxq " O ˜1 pnh n q d ¸" op1q.

	1 pxq `V2 pxq `V3 pxq n pxq| p Var ´p f 1 " f 1 n´1 pxq, p f n´1 pxq ¯ı , ´E " V 1 pxq " E V 2 pxq " E " Var p f 1 n pxq| p f 1 n´1 pxq, p f n´1 pxq ı | p f n´1 pxq ¯ı V 3 pxq " Var ´E " p f 1 n pxq| p f n´1 pxq ı¯. First, in view of (4.3), we can use the following decomposition for the conditional variance (4.20) where and Var ´p f 1 n pxq| p f 1 n´1 pxq, p f n´1 pxq ¯" γ 2 n p f 1 n´1 pxq 2 Var pK n pxqq `γ2 n Var pK 1 n pxqf pX n qq `γ2 n ´σ2 `p f n´1 pxq 2 ¯Var pK 1 n pxqq ´2γ 2 n p f 1 n´1 pxqCov pK n pxq, K 1 n pxqf pX n qq `2γ 2 n p f 1 n´1 pxq p f n´1 pxqCov pK n pxq, K 1 n pxqq ´2γ 2 n p f n´1 pxqCov pK 1 n pxqf pX n q, K 1 n pxqq . ¯2˙`o p1q ˙. Therefore, we obtain from (4.4.1) that V 1 pxq " γ 2 n h ´3 n ˆgpxqξ 2 ˆσ2 `E " ´p f n´1 pxq ´f pxq ¯2˙`o p1q ȧnd Theorem 1 of [6] ensures that V 1 pxq " γ 2 n h ´3 n `gpxqξ 2 σ 2 `op1q ˘. (4.21) In order to compute V 2 , we first observe that E " p f 1 n pxq| p f 1 n´1 pxq, p f n´1 pxq ı " p1 ´γn E rK n pxqsq p f 1 n´1 pxq `γn E rK 1 n pxqf pX n qs ´γn p f n´1 pxqE rK 1 n pxqs and therefore Var ´E " p f 1 n pxq| p f 1 n´1 pxq, p f n´1 pxq ı | p f (4.22) Finally, we can easily obtain V 3 pxq " Var ´γn p f n´1 pxqE rK 1 n pxqs " γ 2 n E rK 1 n pxqs 2 Var ´p f n´1 pxq which, using Theorem 1 of [7], gives us V 3 pxq " o `γ2 n h ´3 n ˘. (4.23) z n pxq " $ & % lim nÑ`8 z n pxq " 0. In view of Lemma 2 of [8], we have lim nÑ`8 (4.42) Furthermore, we have Varpρ n pxqq " E rK 1 n pxqs using once again Theorem 1 of [7], leads to lim nÑ`8 nh 3 n γ n Varpρ n pxqq " 0 which means that It follows from lemma 1 of [7] that V R n pxq " o `γn h ´3 n 125 Using one more time Lemma 1 in [7], we have V H n pxq " o `γn h ´3 n which implies that a nh 3 n H n pxq P ÝÑ0 (4.48) L n pxq ď pnh 3 n q 1`d ℓ 2d Λ n pxq (4.53) First, we note that E " |K 1 n pxq| 2p1`dq ı " 1 h 4p1`dqq n ż R ˇˇˇK 1 ˆx ´xn h n ˙ˇˇˇ2 p1`dq gpx n q dx n " h 3`4d n R |K 1 pyq| 2p1`dq gpx ´yh n q dy " O n ˆ1 h 3`4d ˙(4.58) Λ n pxq " O n ˆ1 n 1`2d h 3`4d ˙. 1 ż E " |µ n pxq| 2p1`dq | F n´1 ‰ " O ˆ1 h 3`4d n ˙(4.64) In view of (4.55) and (4.64), we have once again with lemma 1 in [7] applied to (4.54), γ n (4.45) (4.2.1) that E " ´p f 1 n pxq ´f 1 pxq ¯2 " O `γn h ´3 n ˘. Hence, using (4.16) we obtain that Varpη ˘. Moreover, as soon as α ą 1{7, lim nÑ`8 a nh 3 n ´E " p f 1 n pxq ı (4.51) and (4.8L n pxq " 1 a n n ÿ k"1 E " E " |µ n pxq| 2p1`dq | F n´1 ‰ ď C d `E " |K 1 `E " |ε n K 1 `ˇˇf pxq ´p f n´1 pxq ˇˇ2 p1`dq E " |K 1 n pxq| `f 1 pxqE " |K n pxq| 2p1`dq ‰Ȇ " |R µ We deduce from (4.57), together with (4.59), (4.60), (4.61), (4.62) and (4.63) that n pxq| 2p1`dq | F n´1 ı implies that E " |R µ n pxq| 2p1`dq | F n´1 ı " o n (4.63) ˆ1 h 3`4d ˙. (4.57) 2p1`dq ı n n pxq| 2p1`dq ı which n pxq pf pX n q ´f pxqq | 2p1`dq ‰ ´f 1 pxq ¯" 0 f 1 pxqE " |K n pxq| 2p1`dq ‰˘" O ˆ1 h 1`2d n ˙" o n (4.62) ˆ1 h 3`4d ˙. 2 Var ´f n pxq which, which implies that which, (4.30) together with (4.39),(4.46) and (4.48) leads to where which leads to Consequently, a nh 3 n R n pxq P ÝÑ0. (4.46) " ´p f 1 n´1 pxq ´f 1 pxq ¯2 with E " ´p f 1 n pxq ´f 1 pxq ¯2 " ´E " p f 1 n pxq ı ´f 1 pxq ¯2 `Var ´p f 1 n pxq ¯. Delta n pxq P ÝÑ0. (4.49) Finally, convergence (4.7) immediatly follows from the conjunction of (4.4), (4.30) and (4.49). We are now in the position to prove the asymptotic normality (4.8).First, we obtain from (4.46) and (4.48) that nh 3 a nh 3 n ´p f 1 n pxq ´E " p f 1 n pxq ı¯L ÝÑN ˆ0, 2gpxq ´p1 ´3αq Using similar computation as in (4.58), we also find that (4.50) gpxqξ 2 σ 2 ˙. with E n pxq " γ 1`2d n E " |µ n | 2p1`dq |F n´1 ‰ . (4.55) E " |K 1 n pxq pf pX n q ´f pxqq | 2p1`dq ‰ h 3`4d n " h 1`2d n R |yK 1 pyq| " O ˆ1 h 1`2d n ˙" o n (4.61) ˆ1 h 3`4d ˙. 2p1`dq gpx ´yh n q dy 1 ż ď C f ż n pR n pxq `Hn pxqq Furthermore, since f is Lipschitz continuous, there exists C f such that a Λ n pxq " A n pxq ´2p1`dq n ÿ E " |∆M k | 2p1`dq |F k´1 ‰ E " |ε n K 1 n pxq| 2p1`dq ı " E " |ε n | 2p1`dq ı E " |K 1 n pxq| 2p1`dq ı " O ˙. (4.59) `nh 3 n ˆ1 h 3`4d n k"1 " A n pxq ´2p1`dq n ÿ k"1 A n pxq 2p1`dq γ 2p1`dq n E " ‰ It follows from the latter equation combined with (4.53) that Moreover we also have |µ k | 2p1`dq |F k´1 " p1 ´αn pxqq 2p1`dq Λ n´1 pxq `γ2p1`dq n E " |µ n | 2p1`dq |F n´1 ‰ (4.54) ˇˇf pxq ´p f n´1 pxq ˇˇ2 p1`dq E " |K 1 n pxq| 2p1`dq ı " o ˙a.s. (4.60) L n pxq P ÝÑ0 ˆ1 h 3`4d n which achieves the proof of Theorem (4.8).
	126 127 128 129

n´1 pxq ¯" p1 ´γn E rK n pxqsq 2 Var ´p f 1 n´1 pxq ¯.

w n pxqq " O `γn h ´4 n ˘. P ÝÑ0. R |yh n

  Comme évoqué dans la Remarque 2.3.3, l'étude d'estimateurs par polynômes locaux pourrait aussi être une piste à étudier. Il existe déjà des travaux sur ces types d'estimateurs mais les résultats obtenus ne fonctionnent que dans le cas de noyaux à supports compacts.

		Ñ f 1 pxq	Variance asymptotique	Cas uniforme	α
	NWr J WJ R	p.s. p.s. p.s. P	ξ 2 p1 `3αqgpxq ξ 2 p1 `3αqgpxq pf 2 pxq `σ2 q σ 2 ξ 2 p1 `3αqgpxq pf 2 pxq `σ2 q gpxqξ 2 σ 2 2gpxq ´p1 ´3αq	ξ 2 p1 `3αq pf 2 pxq `σ2 q 1{5 ă α ă 1{3 σ 2 1{5 ă α ă 1{3 p1 `3αq ξ 2 ξ 2 pf 2 pxq `σ2 q 1{5 ă α ă 1{3 p1 `3αq ξ 2 p1 `3αq σ 2 1{7 ă α ă 1{3
	NWd	p.s.	´´´σ 2 ξ 2	1{5 ă α ă 1{3

Remerciements

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Figure 3.9 reveals that the valve velocity distributions for the two periods are significantly different (pvalue ă 0.0001 using a Kolmogorov-Smirnov test). The Quantile-Quantile plot representation given in Figure 3.10 confirms graphically this result. At the highest temperatures for which it was reported that the animals were disturbed and at the limits of their thermal preference [28], the velocity distribution is different from that at the lowest temperatures. We observe on the left side of where A n pxq and M n pxq are given, for all x in r0, 1s, by

M n pxq "

First of all, we focus our attention on the asymptotic behavior of A n pxq. By a Riemann sum approximation argument, we claim that for all x in s0, 1r,

In order to prove convergence (3.17) denote, for any x and y in r0, 1s, a n px, yq " f pyqK 1 ˆx ´y h n ˙.

Since t k ´tk´1 " 1{n, we clearly have

pt k ´tk´1 q a n px, t k q where t 0 " 0. One can easily notice that Chapitre 4

On the asymptotic behavior of the derivative of the Révész estimator

Abstract. This paper is devoted to the nonparametric estimation of the derivative of the regression function in a nonparametric regression model. We implement a procedure based on the derivative of the Révész estimator. We establish the convergence in probability as well as the asymptotic normality for our estimate. Moreover, we compare the asymptotic variance of this estimator to the asymptotic variance of the derivative of the recursive Nadaraya-Watson estimator in order to show in which cases it is better to use derivative of the Révész estimator. We also illustrate the results on simulated data.

Key words. Mathematical statistics ; Nonparametric methods ; Nonparametric regression.

Hence, in view of (4.20),(4.21),(4.22) and (4.23), it follows that

V n pxq " p1 ´2γ n gpxq `opγ n qq V n´1 pxq `γ2 n h ´3 n `gpxqξ 2 σ 2 `op1q ˘.

Therefore, Lemma 1 of [7], ensures that, as soon as 2gpxq ą 1 ´3α, Varp p f 1 n pxqq "

However, if 2gpxq ď 1 ´3α, we have 1 ´3α ą 4α which implies that γ n h ´3 n " oph 4 n q. Thus, Lemma 1 of [7]