Dr Marie Fellmann
email: mariefellmann@hotmail.fr

Loïc Balssa

Elise Clément

Pierre Frey

Alexandre Frontczak

Stéphane Bernardini

Éric Chabannes

Guillaume Guichard

Hugues Bittard

François Kleinclauss

Complications chirurgicales et résultats à long terme des transplantations rénales avec transplants à critères élargis

Complications chirurgicales et résultats à long terme des transplantations rénales avec transplants à critères élargis

Introduction

Depuis une quinzaine d'années, les caractéristiques des transplants provenant de donneurs cadavériques sont de plus en plus larges et ceci en raison des avantages de la transplantation rénale même d'un rein non optimal en comparaison de la dialyse en termes de survie du patient mais aussi de qualité de vie [START_REF] Ojo | Survival in recipients of marginal cadaveric donor kidneys compared with other recipients and wait-listed transplant candidates[END_REF]. Des transplants autrefois refusés au don sont aujourd'hui prélevés et transplantés. Ainsi l'UNOS (United Network of Organ Sharing) a validé la définition de donneurs à critères élargis proposée par Nyberg [START_REF] Nyberg | Improved scoring system to assess adult donors for cadaver renal transplantation[END_REF][START_REF] Metzger | Expanded criteria donors for kidney transplantation[END_REF]. Par définition, la transplantation avec un transplant provenant de donneurs à critères élargis (transplant ECD) est associée à un risque relatif de perte du transplant à trois ans de 1,7 par rapport à un transplant standard [START_REF] Nyberg | Improved scoring system to assess adult donors for cadaver renal transplantation[END_REF]. Cependant compte tenu de la pénurie d'organes et de l'augmentation des délais d'attente avant transplantation, ces transplantations représentaient selon le dernier rapport de l'Agence de la biomédecine plus de 50% des transplantations rénales en France en 2016 [START_REF]Agence de la biomédecine -Le rapport annuel médical et scientifique[END_REF]. À notre connaissance, il a très peu été analysé si ces transplantations avec un transplant à critères élargis étaient associées à une fréquence augmentée de complications chirurgicales.

L'objectif de notre étude était de déterminer si les transplants ECD étaient associés à plus de complications en tenant compte des comorbidités des receveurs. Les receveurs de transplants ECD étaient plus âgés (âge moyen 61,25 +/-13,1 ans versus 46,14 +/-12,2 ; p < 0,001) ; ils présentaient plus de comorbidités (diabète 25,9% versus 13,2, p = 0,005, antécédents cardiovasculaires 18,8% versus 10,3, p = 0,023) et avaient plus de traitements modifiant l'hémostase (anticoagulation curative : 11,7% versus 5,9, p = 0,044, anti-agrégation plaquettaire : 23,9% versus 13,2, p = 0,016).

Matériels et Méthodes

Nous

Il n'y avait pas de différence statistiquement significative entre les deux groupes concernant le tabagisme actif, le type de dialyse ou le temps d'ischémie froide. Les caractéristiques épidémiologiques de la population sont présentées Tableau 1.

Les patients recevant un transplant ECD présentaient plus de reprise de fonction retardée du transplant (23,3% vs 12,4 p = 0,01), restaient hospitalisés plus longtemps 12,1 jours +/-10,4 versus 10,3 +/-11,6 jours (p = 0,003). Ils ont nécessité plus de transfusions 46,1% vs 28,2 (p = 0,007). Ils ont présenté plus de complications chirurgicales (52,5% versus 39,9, p = 0,082), notamment à type d'hématomes (18,3% versus 10,6 p = 0,045) et plus de sténose de l'anastomose uréterovésicale (4,6% versus 0,7, p = 0,011). Ils présentaient plus de complications de Clavien IVa : 16,2% versus 9,2, p = 0,048.

Le CCI (Comprehensive Complication Index) était à 22,1contre 8,9 ; p = 0,002.

Les résultats concernant la période postopératoire sont présentés Tableau 2.

Le suivi médian des patients était de 61,4 mois [22,4 -93,89].

Les patients recevant des transplants ECD avaient un plus faible taux de survie globale et taux de survie du transplant comme le montrent les courbes de survie figure 1 Si en analyse univariée, les transplants ECD semblaient présenter plus de complications chirurgicales comme dans l'étude de Battaglia [START_REF] Battaglia | Medical and surgical complications after kidney transplantation from « suboptimal donors »: one centre's experience[END_REF], ceci n'était pas retrouvé dans notre analyse multivariée. Les comorbidités du receveur étaient des facteurs de confusion majeurs qui n'étaient pas pris en compte chez Battaglia. Pour les complications chirurgicales et plus particulièrement les hématomes : le facteur de confusion majeur était le traitement anticoagulant qui était plus présent chez les receveurs de transplants ECD.

Ceci est probablement dû au fait que les transplants ECD sont plus transplantés à des receveurs plus âgés, présentant plus de comorbidités [START_REF] Metzger | Expanded criteria donors for kidney transplantation[END_REF][START_REF] Sung | Impact of the expanded criteria donor allocation system on candidates for and recipients of expanded criteria donor kidneys[END_REF] ce qui était confirmé dans notre étude : Tableau1. Cette distribution non aléatoire des transplants par l'agence de la biomédecine fait suite à des études qui ont montré que les transplants ECD ne devaient pas être transplantés chez les patients jeunes [START_REF] Schold | Which Renal Transplant Candidates Should Accept Marginal Kidneys in Exchange for a Shorter Waiting Time on Dialysis?[END_REF]. Des études montrent que la transplantation de transplants ECD est justifiable chez les patients avec une faible espérance de vie [START_REF] Rouchi | When is Transplantation with a « Marginal Kidney » Justifiable?[END_REF]. Si les patients avec des transplants ECD présentent plus de complications chirurgicales, il semble que ce soit simplement car ils ont plus de comorbodités et non car leur transplant est ECD. L'absence de différence significative en analyse multivariée est peut être aussi liée au fait que les événements de type hématomes ou sténose urétrale sont trop rares et donc nos effectifs trop faibles pour mettre en évidence une différence statistiquement significative entre nos deux groupes.

Dans notre étude, la reprise de fonction retardée du transplant était plus fréquente avec les transplants ECD et ce résultat était confirmé en analyse multivariée. Ceci est en accord avec la littérature [START_REF] Pérez-Sáez | Early outcomes of kidney transplantation from elderly donors after circulatory death (GEODAS study)[END_REF][START_REF] Bozkurt | Marginal Donors in Renal Transplantation[END_REF] et serait lié au fait que les lésions d'ischémie-reperfusion sont plus importantes chez les transplants ECD [START_REF] Szwarc | Delayed graft function: a frequent but still unsolved problem in renal transplantation[END_REF][START_REF] Barrou | Ischémie-reperfusion. Liquides de conservation et machines de perfusion en transplantation rénale[END_REF]. Pour tenter de limiter ces lésions, mettre ces reins sur machine à perfuser a montré un bénéfice majeur [START_REF] De Deken | Hypothermic machine perfusion in kidney transplantation[END_REF][START_REF] Vaziri | Analysis of machine perfusion benefits in kidney grafts: a preclinical study[END_REF][START_REF] Savoye | Evaluation of outcomes in renal transplantation with hypothermic machine perfusion for the preservation of kidneys from expanded criteria donors[END_REF]. Ainsi, il y aurait une diminution significative de reprise de fonction retardée du transplant. La survie des transplants ECD à un an serait améliorée de 12% [START_REF] Treckmann | Machine perfusion versus cold storage for preservation of kidneys from expanded criteria donors after brain death[END_REF], ils rejoindraient ainsi les transplants standards. Ceci doit encore être étudié car une étude toute récente mais avec des effectifs probablement insuffisants ne montrait pas de différence significative ni sur le taux de reprise de fonction retardée du transplant ni sur la survie des transplants avec un suivi de 6 ans [START_REF] Basu | Outcomes of Deceased Donor Kidney Transplantation Using Expanded Criteria Donor Kidneys Following Pulsatile Preservation[END_REF].

Quoi qu'il en soit, en France, l'agence de la biomédecine a mis en place un financement dédié à la prise en charge des machines à perfuser dans le cadre de la préservation des transplants ECD [START_REF] Barrou | Ischémie-reperfusion. Liquides de conservation et machines de perfusion en transplantation rénale[END_REF].

Nos résultats confirment que les transplants ECD ont une moins bonne survie sur le long terme et c'est ainsi que les transplants ECD ont été définis initialement [START_REF] Borda | Functional and morphological changes in kidneys from marginal donors[END_REF][START_REF] Chapman | Marginal kidneys for transplantation[END_REF]. Pour compenser ce risque de perte du transplant, des précautions complémentaires autres que la mise sur machines à perfuser peuvent être mises en oeuvre sur la période péri-opératoire comme réduire le temps d'ischémie froide [START_REF] Krüger | Early experience with the ET Senior Program « Old For Old »; better to be number one?[END_REF] ou du moins le ramener en dessous du seuil de 800 minutes (13 heures) pour les transplants ECD [START_REF] Denecke | Reduction of Cold Ischemia Time and Anastomosis Time Correlates with Lower Delayed Graft Function Rates Following Transplantation of Marginal Kidneys[END_REF], sélectionner les receveurs, adapter les immunosuppresseurs, réaliser des bi-transplantations [START_REF] Audard | Renal transplantation from extended criteria cadaveric donors: problems and perspectives overview[END_REF][START_REF] Remuzzi | Early experience with dual kidney transplantation in adults using expanded donor criteria[END_REF], ou encore se servir d'index comme le KDPI (Kidney donor profile index), prédictif de la longévité du greffon [START_REF] Gupta | KDPI score is a strong predictor of future graft function: Moderate KDPI (35 -85) and high KDPI (> 85) grafts yield similar graft function and survival[END_REF] afin que celui ci soit transplanté à la personne la mieux adaptée et ainsi optimiser les résultats de la transplantation. Certaines équipes proposent également la réalisation de scores histologiques sur des biopsies de reins ECD pour déterminer si ils sont éligibles à la transplantation [START_REF] Carta | Impact of the pretransplant histological score on 3-year graft outcomes of kidneys from marginal donors: a single-centre study[END_REF].

Il est à noter que pour les donneurs vivants, même avec des critères plus sévères que ceux validés par l'UNOS, il n'y a pas de différence sur la survie des transplants à 5 ans [START_REF] Oikawa | Safety and Effectiveness of Marginal Donor in Living Kidney Transplantation[END_REF]. C'est pour cela que nous avons décidé de les exclure de notre analyse.

Notre étude comporte certaines limites : elle est rétrospective, monocentrique avec une période d'inclusion de 11 ans ce qui est long. Cependant il s'agit d'une série historique qui nous permet d'avoir un effectif important et une analyse la plus exacte possible des résultats de la transplantation avec transplants ECD juste avant la démocratisation des machines à perfuser en France et avant l'utilisation de transplants prélevés sur des donneurs à coeur arrêté.

Il serait très intéressant de refaire ce type d'étude avec tous les transplants ECD mis sur machine ces dernières années surtout en terme de survie du transplant.

Conclusion

La transplantation de transplants ECD n'entraîne pas plus de complications que celle réalisée à partir de transplants standards mais est associée à une reprise de fonction retardée du transplant. Connaissant la fragilité de ces transplants concernant le capital néphronique et la survie du transplant à long terme, il convient de les proposer à des patients plus âgés et de favoriser leur longévité en limitant les lésions d'ischémie-reperfusion. Une évaluation de l'utilisation de machines à perfusion dans cette indication est souhaitable.

  avons réalisé une étude rétrospective incluant 515 patients consécutifs ayant bénéficié d'une transplantation rénale dans notre centre entre le 1 er janvier 2005 et le 31 décembre 2016. 9 d'entre eux ont été exclus du fait d'une transplantation combinée foie-rein et 36 d'entre eux du fait d'un rein provenant de donneur vivant. Les transplants étaient tous issus de donneur en état de mort encéphalique, aucun ne provenait d'un donneur à coeur arrêté. La population a été répartie en deux groupes : les patients ayant reçu un transplant de donneur à critères élargis (ECD) et ceux de donneurs standards. La définition d'un transplant à critères élargis était celle validée par l'UNOS ( âge supérieur à 60 ans, ou âge entre 50 et 59 ans associé à deux des trois critères suivants : antécédents d'hypertension artérielle, décès de cause cérébrovasculaire, créatininémie supérieure à 1,5 mg/dl au moment du prélèvement) [3]. Aucun transplant n'a été conservé sur machine à perfusion. Les caractéristiques des deux populations ont été rapportées et analysées : -Concernant les receveurs : l'âge, le sexe, les comorbidités du receveur : diabète, antécédents cardio-vasculaire (cardiopathie ischémique ou accident vasculaire cérébral ou artériopathie oblitérante des membre inférieurs stade 2 minimum : douleur à la marche selon la classification de Leriche-Fontaine), tabagisme actif, traitements antiagrégants et anticoagulants durant la période periopératoire (anticoagulation curative reprise ou introduite avant le troisième jour postopératoire, antiagrégation non stoppée en périopératoire), le type et la durée de dialyse préopératoire. -Concernant les transplants et la transplantation : temps d'ischémie froide, la présence d'athérome sur les artères, les variations anatomiques (vasculaires et urétérales), la réalisation d'une anastomose artérielle sur patch ou plein canal. Tous les évènements survenus dans la période post-opératoire précoce, (< 30 jours suivant l'intervention) ont été analysés: durée de séjour, reprise de fonction retardée du transplant définie par la nécessité d'un recours à la dialyse en postopératoire [5], infections du site opératoire définies comme une infection de paroi ou une infection urinaire, les complications chirurgicales par type : Vasculaire (thrombose artérielle, thrombose veineuse, hémorragie) ; urinaire (sténose, fuite, hématurie) ; lymphatique (lymphocèle) ; les complications de paroi (désunion). Les complications post-opératoires ont été classées selon la classification de Clavien et Dindo [6] et selon le Comprehensive Complication Index [7] puis comparées. La survie du patient et celle du transplant ont été analysées dans chaque population et comparées. La survie du transplant était définie par la période entre la transplantation et le retour définitif à la dialyse ou le décès du patient. Analyse statistique Les données quantitatives sont présentées comme moyenne +/-déviation standard si leur distribution est normale et comme médiane [interquartile] si distribuées non normalement. La distribution des variables quantitatives était calculée avec le test de Kolmogorov-Smirnov. Les variables qualitatives sont présentées comme effectif (pourcentage). Le test du χ2 était utilisé pour comparer des variables catégorielles et le test de Mann Withney pour les variables quantitatives. Les résultats étaient considérés comme significatif si p était inférieur à 0,05. Les facteurs significatifs dans l'analyse univariée ainsi que ceux reconnus dans la littérature étaient inclus dans une régression logistique pas à pas descendante (Stepwise) pour l'analyse multivariée. Les résultats finaux des modèles multivariés sont présentés par le Hazard Ratio et l'intervalle de confiance à 95%. Les analyses de survie ont été réalisées avec les courbes de Kaplan Meier et comparées par une analyse de Cox. Toutes ces analyses étaient réalisées avec le logiciel Medcalc version 11.2.1.0 (Medcalc, Mariakerke, Belgium). Résultats 470 patients ont été inclus dans l'analyse finale (âge moyen : 52,49 ans+/-12,6, 319 (67,9%) de sexe masculin). 273 (58,1%) transplants provenaient de donneurs standards et 197 (41,9%) de donneurs à critères élargis.

  et 2, p =0,005 et 0,001 respectivement. Les transplants ECD étaient associés en analyse multivariée à une reprise de fonction retardée du transplant (Tableau 3) : HR = 2,1 (1,1 -3,68), p = 0,008. L'obésité HR = 2,53 (1,32 -4,83), p = 0,005 et le temps d'ischémie froide HR = 2,18 (1,25 -3,47), p = 0,005 étaient également des facteurs indépendants associés à la reprise de fonction retardée du transplant. Les transplants ECD n'étaient pas associés à une augmentation de la survenue de complications chirurgicales en analyse multivariée (Tableau 4). Seule la présence d'un traitement anticoagulant en période péri opératoire était associée à une fréquence plus élevée de complications chirurgicales : HR = 3,27 (1,57 -6,63), p = 0,003. Les transplants ECD n'étaient pas associés à la survenue d'hématome (Tableau 5) : HR = 1,56 (0,83 -2,85), p = 0,231. Seuls les antécédents cardiovasculaires et l'anticoagulation curative l'étaient : HR = 2,24 (1,15 -4,74), p = 0,034 et HR = 3,61 (1,77 -7,41), p < 0,001. L'analyse multivariée sur la durée de séjour et les sténoses urinaires ne faisaient pas ressortir les transplants ECD comme facteur favorisant. Les analyses de Cox réalisées sur la survie des patients et des transplants sont présentées Tableaux 6 et 7. Les transplants ECD étaient statistiquement significativement associés à une moindre survie des transplants : HR = 1,81 (1,1 -2,98), p = 0,029 mais n'étaient pas associés à une moindre survie des patients HR = 1,12 (0,58 -2,17), p = 0,763. Discussion Dans notre étude, les transplants ECD n'étaient pas associés à une survenue plus fréquente de complications postopératoires en dehors d'une reprise de fonction retardée du transplant. Ils étaient également associés à une moindre survie des transplants.

Figure 1 :

 1 Figure 1: Survie des patients

Figure 2 :

 2 Figure 2: Survie des transplants

Table 6 :

 6 Cox analysis : Graft survival

	Tableau 1 : Données démographiques de la population
	Table 1 : Population
	Tableau 1: Donnée démographiques de la population
		Donneur standards	Donneurs à critères élargis	p
	Transplants	273 (58,1)	197 (41,9)	0,005
	Sexe masculin	179 (65,6)	140 (71,1)	0,993
	Age (ans)	46,14 +/-12,2	61,25 +/-13,1	<0,001
	IMC (kg/m2)	23,8 [21,3-26,7]	24,4 [22-27,6]	0,288
	Diabète	36 (13,2)	51 (25,9)	0,005
	Antécédents cardiovasculaires	28 (10,3)	37 (18,8)	0,023
	Tabagisme actif	87 (29,1)	52 (26)	0,512
	Anticoagulation curative≤J3	16 (5,9)	23 (11,7)	0,044
	Antiagrégation plaquettaire	36 (13,2)	47 (23,9)	0,016
	Dialyse
	Hémodialyse	179 (65,6)	140 (71,1)	0,608
	Dialyse péritonéale	62 (22,7)	46 (23,4)	0,914
	Greffe préemptive	32 (11,7)	12 (6,1)	0,076
	Durée de dialyse (jours)	571 [229-1146]	815 [431-1203]	<0,001
	Informations sur le greffon
	Durée d'ischémie froide (minutes)	919 [709-1118]	914 [779-1159]	0,343
	Artères athéromateuses	49 (17,9)	91 (46,2)	<0,001
	Patch artériel	195 (71,4)	148 (75,1)	0,773
	Anomalie vascularisation artérielle	54 (19,8)	41 (20,8)	0,821
	Anomalie vascularisation veineuse	10 (3,7)	5 (2,5)	0,601
	Anomalie uretère	6 (2,2)	2 (1)	0,478
	Variables présentées comme Nombre (%), Médiane[Interquartile],
	Moyenne +/-déviation standard
	IMC: Indice de masse corporelle

Table 7 :

 7 Cox analysis : Patients survival

	Tableau 7: Régression de Cox: survie des patients
		HR	IC 95%	p
	Age receveur	1,09	1,03 -1,14	0,083
	Diabète	1,51	0,75 -3,03	0,286
	Antécédents cardiovasculaires	2,72	1,38 -5,23	0,006
	Donneur à critères élargis	1,12	0,58 -2,17	0,763
	HR = Hazard Ratio, IC = Intervalle de confiance

Tableau 2 : Analyse univariée de la période post-opératoire


domain: math stat

text:
Gabriela Ciuperca
email: gabriela.ciuperca@univ-lyon1.fr

Variable selection in high-dimensional linear model with possibly asymmetric errors

Keywords: adaptive LASSO, expectile, high-dimension, oracle properties

In many application areas, the problem of the automatic variable selection in a linear model with asymmetric errors is encountered, when the number of explanatory variables diverges with the sample size. For this high-dimensional model, the penalized least squares method is not appropriate and the quantile framework makes the inference more difficult because of the non differentiability of the loss function. An estimation method by penalizing the expectile process with an adaptive LASSO penalty is proposed and studied. Two cases are considered: first with the number of model parameters is assumed to be much smaller than the sample size and afterwards it could be of the same order; the two cases being distinct by the adaptive penalties considered. For each case, the rate convergence is obtained and the oracle properties of the adaptive LASSO expectile estimator are established. The proposed estimators are evaluated through Monte Carlo simulations and compared with the adaptive LASSO quantile estimator. The proposed estimation method is also applied to real data in genetics.

Introduction

The focus of the present paper is to better detect significant variables in a linear model, with the possibility that the number of explanatory variables varies with the sample size and when the error distribution is asymmetrical. For this type of law, the use of the least squares(LS) estimation method is not appropriate because of the estimator's accuracy (see [START_REF] Liao | Penalized expectile regression: an alternative to penalized quantile regression[END_REF]). One possibility would be to use the quantile method, but it has the disadvantage that the loss function is not derivable, which complicates the theoretical study but also the computational methods. A very interesting possibility is to consider the expectile method, introduced by [START_REF] Newey | Asymmetric least squares estimation and testing[END_REF], under assumption that the first moments of ε exist. This method has the advantage over the quantile model that the loss function is differentiable, which makes the theoretical study more amenable and considerably facilitates the numerical computation (see also [START_REF] Gu | High-dimensional generalizations of asymmetric least squares regression and their applications[END_REF]). Readers can find in [START_REF] Newey | Asymmetric least squares estimation and testing[END_REF], [START_REF] Schnabel | Optimal expectile smoothing[END_REF] the theoretical and computational advantages of expectile method over quantile regression. Two of these advantages are: the efficiency of the expectile estimator and the fact that its asymptotic variance can be calculated without going via the density of the model errors. The advantage of the quantile estimation method is that it is more robust to outliers than the expectile method, the last being also more sensitive to the extreme values in the response variable (see also [START_REF] Zhang | A continuous threshold expectile model[END_REF]). In application fields (genetics, chemistry, biology, industry, finance), with the development in recent years of storage and/or measurement tools, we are confronted to study the influence of a very large number of variables on a studied process. That is why, we consider in the present work the following linear model:

Y i = X t i β + ε i , i = 1, • • • , n, (1)
with the vector parameter β ∈ R p and β 0 its true value (unknown). The size p of β 0 can depend on n but the components β 0 j don't depend on n, for any j = 1, • • • , p. The vector X t i = (X i1 , • • • , X ip ) t contains the values of the p explanatory deterministic variables and Y i the values of response variable for observation i. The values (Y i , X i ) are known for any i = 1, • • • , n. Throughout the paper, all vectors are column. If p is very large, in order to find the explanatory variables that significantly influence the response variable Y , an automatic selection should be made without performing hypothesis tests. Concerning the hypothesis testing of coefficients in high dimensional linear regression model, a lot of progress has been made in recent years. For Gaussian errors, with possibility that the number of variables exceeds the sample size, [START_REF] Zhang | Confidence intervals for low dimensional parameters in high dimensional linear models[END_REF] propose a method for constructing confidence interval for individual coefficients, which can be used to select variables after proper thresholding. [START_REF] Dezeure | High-Dimensional Inference: Confidence Intervals, p -Values and R -Software hdi[END_REF] present a review of frequentist methods for constructing p-values and confidence intervals in a high dimensional linear model. Always for this model, with zero mean errors, [START_REF] Dezeure | High-dimensional simultaneous inference with the bootstrap[END_REF], [START_REF] Shah | Goodness-of-fit tests for high dimensional linear models[END_REF] propose bootstrap methodology.

For model (1), let the index set of the non-null true parameters,

A ≡ j ∈ {1, • • • , p}; β 0 j = 0 .
Since β 0 is unknown then, the set A is also unknown. We assume, without reducing generality, that A = {1, • • • , p 0 } and its complementary set is A c = {p 0 + 1, • • • , p}, with p 0 ≤ p. Hence the first p 0 explanatory variables have a significant influence on the response variable and the last p -p 0 variables are irrelevant. Thus, the true parameter vector can be written as β 0 = β 0 A , β 0 A c = β 0 A , 0 p-p 0 , with 0 p-p 0 a (p -p 0 )-vector with all components zero. The number p 0 of the nonzero coefficients can depend on n.

For a vector β we use the notational convention β A for its subvector containing the corresponding components of A. For i = 1, • • • , n, we denote by X i,A the p 0 -vector with the components X ij , j = 1, • • • , p 0 . We also use the notation |A| or Card(A) for the cardinality of A. In order to find the elements of A, one of the most used techniques is the adaptive LASSO method, introduced for p fixed by [START_REF] Zou | The adaptive Lasso and its oracle properties[END_REF] by penalizing the squares sum with an weighted L 1 penalty. This type of parameter estimator is interesting if it satisfies the oracle properties, i.e. the two following properties occur:

• sparsity of estimation: the non-zero parameters are estimated as non-zero and the null parameters are shrunk directly as 0, with a probability converging to 1 when n → ∞;

• asymptotic normality of non-zero parameter estimators.

In order to distinguish between different types of adaptive LASSO estimators, we will use the term "adaptive LASSO LS-estimator" to refer to the minimizer of the LS sum penalized with adaptive LASSO. Let us give some papers from very rich literature that consider the adaptive LASSO LSestimator when p depends on n: [START_REF] Huang | Adaptive Lasso for sparse high-dimensional regression models[END_REF], [START_REF] Wang | Parametric component detection and variable selection in varying-coefficient partially linear models[END_REF], [START_REF] Yang | Nonnegative adaptive lasso for ultra-high dimensional regression models and a two-stage method applied in financial modeling[END_REF]. If the moments of the errors do not exist or the distribution of ε presents outliers, then the LS framework is not appropriate. One possibility is to consider the quantile model with the adaptive LASSO penalty. The recent literature is also very rich: [START_REF] Fan | Adaptive robust variable selection[END_REF], [START_REF] Kaul | Weighted l 1 -penalized corrected quantile regression for high dimensional measurement error models[END_REF], [START_REF] Tang | Variable selection in high-dimensional quantile varying coefficient models[END_REF], [START_REF] Ciuperca | Adaptive group LASSO selection in quantile models[END_REF], [START_REF] Zheng | Adaptive penalized quantile regression for high dimensional data[END_REF], [START_REF] Zheng | Globally adaptive quantile regression with ultra-high dimensional data[END_REF], to give just a few examples. As stated before, the non-differentiability of the loss function for quantile method complicates the theoretical study and its computational implementation, which is a very important aspect in highdimensionality. Let us mention another work of [START_REF] Fan | Estimation of high dimensional mean regression in the absence of symmetry and light tail assumptions[END_REF] which is also devoted to the high dimensional regression in absence of symmetry of the model errors and which proposes a penalized Huber loss but in which the sparsity of the robust approximate LASSO estimator is not studied. The robust approximate LASSO estimator is consistent with the same convergence rate as the optimal rate under the light tail situation. In the present paper we consider the expectile loss function for a high-dimensional model. In order to introduce the expectile method, for a fixed τ ∈ (0, 1), let us consider the function ρ τ (.) of the form

ρ τ (x) = |τ -1 1 x<0 |x 2 , with x ∈ R.
For the error and the design of model (1) we make the following basic assumptions. The errors ε i satisfy the following assumption:

(A1) (ε i ) 1 i n are i.i.d. such that E[ε 4 i ] < ∞ and E[ε τ 1 1 ε>0 + (1 -τ )1 1 ε<0 ] = 0, that is its τ -th expectile is zero: E[ρ τ (ε)] = 0.
While, the design (X i ) 1 i n satisfies the following assumption:

(A2) there exists two positive constants m 0 , M 0 such that, 0 < m 0 ≤ µ min n -1 n i=1 X i X t i ≤ µ max n -1 n i=1 X i X t i ≤ M 0 < ∞.
For a positive definite matrix, we denote by µ min (.) and µ max (.) its largest and smallest eigenvalues, respectively. Let us consider ε the generic variable for the sequence (ε i ) 1 i n . Assumption (A1) is commonly required for the expectile models, see Zhao et al. (2018), [START_REF] Gu | High-dimensional generalizations of asymmetric least squares regression and their applications[END_REF], [START_REF] Liao | Penalized expectile regression: an alternative to penalized quantile regression[END_REF], [START_REF] Newey | Asymmetric least squares estimation and testing[END_REF], while assumption (A2) is standard in linear model for the parameter identifiability (considered also by [START_REF] Gao | Asymptotic analysis of high-dimensional LAD regression with Lasso[END_REF], [START_REF] Wang | Adaptive Lasso estimators for ultrahigh dimensional generalized linear models[END_REF], [START_REF] Fan | Estimation of high dimensional mean regression in the absence of symmetry and light tail assumptions[END_REF], [START_REF] Zou | On the adaptive elastic-net with a diverging number of parameters[END_REF]). Other assumptions will be stated about design in the following two sections, depending on the size p which varies in turn with n.

Quite in general, it is wise to use the expectile method when the moments of ε exist but its distribution is asymmetric. For τ = 0.5, we get the classical method of least squares. For model (1), consider the expectile process

Q n (β) ≡ n i=1 ρ τ (Y i -X t i β),
and the one with LASSO adaptive penalty:

R n (β) ≡ n i=1 ρ τ (Y i -X t i β) + nλ n p j=1 ω n,j |β j |. (2)
The adaptive weights ω n,j will be defined later depending on the size of p in respect to n. The tuning parameter λ n is a positive deterministic sequence which together with ω n,j controls the overall model complexity. Hence, we should choose λ n and ω n,j such that nλ n ω n,j P -→ n→∞ 0 for non-null parameters and nλ n ω n,j P -→ n→∞ ∞ for null coefficients. In order to automatically detect the null and non-zero components of β, we proceed in a similar way as for the adaptive LASSO LS-estimation introduced by [START_REF] Zou | The adaptive Lasso and its oracle properties[END_REF], and we consider the adaptive LASSO expectile estimator of β:

β n ≡ arg min β∈R R n (β).
(3)

The components of

β n are β n = β n,1 , • • • , β n,p .
Similarly to A, let's define the index set:

A n ≡ {j ∈ {1, • • • , p}; β n,j = 0},
with the non-zero components of the adaptive LASSO expectile estimator. The estimator β n will satisfy the oracle properties if:

• sparsity: lim n→∞ P A = A n = 1.

• asymptotic normality: for any vector u ∈ R p 0 with bounded norm, we have that: √ n(u t Υ -1 n,A u) -1/2 u t ( β n -β 0 ) A converges in distribution to a zero-mean Gaussian law, with the p 0 -squared matrix:

Υ n,A ≡ n -1 n i=1 X i,A X t i,A .
For p fixed, the properties of the estimator β n have been studied by [START_REF] Liao | Penalized expectile regression: an alternative to penalized quantile regression[END_REF] where it is shown that the convergence rate of β n towards β 0 is of order n -1/2 and that β n satisfies the oracle properties. The case where p is fixed has also been studied by Zhao and Zhang (2018), where consider a penalized linear expectile regression with SCAD penalty function and obtain a n -1/2 -consistent estimator with oracle properties. In the present paper we assume that p depends on n, more precisely, p = O(n c ), with the constant c ∈ [0, 1].

The size p 0 and the set A can also depend on n.

The case when p depends on n was also considered in Zhao et al. (2018) by considering the SCAD penalty for the expectile process. They propose an algorithm that converges, with probability converging to one as n → ∞, to the oracle estimator after several iterations. Always for p depending on n, for (ε i ) sub-Gaussian errors, [START_REF] Gu | High-dimensional generalizations of asymmetric least squares regression and their applications[END_REF] penalize the expectile process with LASSO or nonconvex penalties. They find the convergence rate of the penalized estimator, propose an algorithm for finding this estimator and implement the algorithm in the R language in package SALES. The paper of [START_REF] Spiegel | Model selection in semiparametric expectile regression[END_REF] introduces several approaches depending on selection criteria and shrinkage methods to perform model selection in semiparametric expectile regression.

Let us give some general notations. For a vector v, we denote its transpose by v t , by v 1 , v 2 and v ∞ the L 1 , L 2 , L ∞ norms, respectively. The number p of the explanatory variables and p 0 of the significant variables can depend on n, but for convenience, we do not write the subscript n. Throughout the paper, C denotes a positive generic constant that does not dependent on n, which value may differ from one formula to another.

In order to study the properties of the adapted LASSO expectile estimator β n , we introduce the following functions, using the same notations as in [START_REF] Liao | Penalized expectile regression: an alternative to penalized quantile regression[END_REF]:

g τ (x) ≡ ρ τ (x -t)| t=0 = 2τ x1 1 x≥0 + 2(1 -τ )x1 1 x<0 , h τ (x) ≡ ρ τ (x -t) t=0 = 2τ 1 1 x≥0 + 2(1 -τ )1 1 x<0
The paper is organized as follow. In Section 2 we study the asymptotic behaviour of the adaptive LASSO expectile estimator when p = O(n c ), with 0 ≤ c < 1/2. We obtain the convergence rate of the β n and the oracle properties. A similar study is realized in Section 3, when c ∈ [1/2, 1]. In Section 4, a simulation study and an application to real data are presented. All the proofs are relegated in Section 5.

Case c < 1/2, parameter number less than the sample size

In this section we study the asymptotic behavior of the adaptive LASSO expectile estimator when the number p of model parameter is p = O(n c ), with 0 ≤ c < 1/2. If c = 0, that is p fixed, then we get the particular case studied by [START_REF] Liao | Penalized expectile regression: an alternative to penalized quantile regression[END_REF].

An additional assumption to (A2) on the design is requested:

(A3) p 1/2 n -1/2 max 1 i n X i 2 -→ n→∞ 0.
The upper bound 1/2 for c is deduced taking into account assumptions (A2) and (A3). Since tr

X t i X i = tr X i X t i = X i 2 2 ≤ max 1 i n X i 2 2
, taking into account assumption (A2) we have that max 1 i n X i 2 2 ≥ Cp. This last relation, together with assumption (A3) involve pn -1/2 → 0, as n → ∞, from where c < 1/2. Because p < n, the regression parameters are identifiable and we can calculate the expectile estimator:

β n ≡ arg min β∈R Q n (β), the components of β n being β n = β n,1 , • • • , β n,p
. This estimator will intervene in the adaptive weight of the penalty,

ω n,j = | β n,j | -γ , for j = 1, • • • , p. (4)
Conditions on the constant γ > 0 will be specified in Theorem 2.2. By the following theorem we obtain the convergence rate of the expectile and adaptive LASSO expectile estimators. We obtain that the convergence rate depends on the size p of the vector β.

Theorem 2.1 Under assumptions (A1), (A2) and (A3) we have:

(i) β n -β 0 2 = O P p n . (ii) If the tuning parameter sequence (λ n ) n∈N satisfies p 1/2 0 n (1-c)/2 λ n → 0, as n → ∞, then, β n -β 0 2 = O P p n .
Theorem 2.1 provides that the expectile and adaptive LASSO expectile estimators have the same convergence rate. Concerning the adaptive LASSO expectile estimator, the same convergence rate has been obtained for other adaptive LASSO estimators: by the likelihood method for a generalized linear model when p < n in [START_REF] Wang | Adaptive Lasso estimators for ultrahigh dimensional generalized linear models[END_REF], by the least squares approximation method in [START_REF] Leng | Least squares approximation with a diverging number of parameters[END_REF]. By the following theorem, considering a supplementary condition on λ n , c and γ, in addition to that considered for the convergence rate in Theorem 2.1, we show that the adaptive LASSO expectile estimator β n satisfies the oracle properties. If τ = 0.5, that is, for the adaptive LASSO LS-estimator, the variance of the normal limit law is the variance of ε. In fact, we obtained for τ = 0.5, the same asymptotic normality as in [START_REF] Zou | On the adaptive elastic-net with a diverging number of parameters[END_REF].

Theorem 2.2 Suppose that assumptions (A1), (A2) and (A3) hold and that the tuning parameter satisfies

λ n n (1-c)(1+γ)/2 → ∞, p 1/2 0 n (1-c)/2 λ n → 0, as n → ∞. Then: (i) P A n = A → 1, for n → ∞.
(ii) For any vector u of size p 0 such that u 2 = 1, we have:

n 1/2 (u t Υ -1 n,A u) -1/2 u t ( β n - β 0 ) A L -→ n→∞ N 0, Var [gτ (ε)] E 2 [hτ (ε)] .
The convergence rate of β n does not depend on the power γ, but otherwise, for holding the oracle properties, the choice of γ is very important. Concerning the suppositions and results stated in Theorem 2.2, let's make some remarks on the regularization parameter λ n , the constants c, γ and the sizes p 0 , p.

Remark 2.1 1) If p 0 = O(p), then if we want that λ n n (1-c)(1+γ)/2 → ∞ and λ n p 1/2 0 n (1-c)/2 → 0 occur, we must choose the constant γ and sequence (λ n ) such that: γ > c/(1 -c) and n -1/2 λ n -→ n→∞ 0.
2) If p 0 = O(1), we must choose the constant γ > 0 and the tuning parameter such that [START_REF] Liao | Penalized expectile regression: an alternative to penalized quantile regression[END_REF] for a linear model with p fixed. We also find the same variance of Gaussian distribution in [START_REF] Liao | Penalized expectile regression: an alternative to penalized quantile regression[END_REF].

n (1-c)/2 λ n -→ n→∞ 0. 3) If we want that λ n p 1/2 0 n (1-c)/2 → 0 holds, it is necessary that λ n → 0, as n → ∞. 4) If c = 0 then the conditions on λ n become: n 1/2 λ n → 0 and n (1+γ)/2 λ n → ∞, conditions considered by
Remark 2.2 If ε ∼ N (0, σ 2 ), then τ = 1/2 and the variance of the Normal limit law of the adaptive LASSO expectile estimators for the non-zero parameters, given by Theorem 2.2(ii), is equal to σ 2 . The analogous result for the adaptive LASSO quantile estimator obtained by [START_REF] Ciuperca | Adaptive group LASSO selection in quantile models[END_REF] gives a variance of πσ 2 /2. Thus, adaptive LASSO expectile estimator is more efficient than the adaptive LASSO quantile estimator.

Case c ∈ [1/2, 1]

In this section, after we propose an adaptive weight, we study the asymptotic behavior of the estimator β n when the number of regressors p = O(n c ), with 1/2 ≤ c ≤ 1, with the possibility that p is of the same order as the number of observations, with p ≤ n.

Instead of assumption (A3), we consider:

(A4) There exists a constant M > 0 such that max 1 i n X i ∞ < M .
The same assumption (A4) was considered for a generalized linear model in [START_REF] Wang | Adaptive Lasso estimators for ultrahigh dimensional generalized linear models[END_REF] where the adaptive LASSO likelihood method is proposed.

In addition to the weight ω n,j given by relation (4), in this section we also propose:

ω n,j = min(| ∨ β n,j | -γ , n 1/2 ), (5) with ∨
β n,j an estimator of β 0 j consistent with a n → 0 the convergence rate:

∨ β n -β 0 2 = O P (a n ).
These weights are proposed because when the estimator ∨ β n,j takes the value 0, then we consider n 1/2 as adaptive weight. An example of such estimator is the LASSO expectile estimator, proposed by [START_REF] Gu | High-dimensional generalizations of asymmetric least squares regression and their applications[END_REF], defined as: arg min [START_REF] Gu | High-dimensional generalizations of asymmetric least squares regression and their applications[END_REF]).

β∈R p n -1 n i=1 ρ τ (Y i -X t i β) + ν n β 1 , with the deterministic sequence ν n ∈ (0, ∞), ν n → 0 as n → ∞. If ε i is sub-Gaussian and E[g(ε)] = 0, under our assumptions (A2), (A4), if κ = inf d∈C Xd 2 2 nd 2 2 ∈ (0, ∞), with X the matrix n × p of design and the set C ≡ {d ∈ R p ; d A c 1 ≤ 3 d A 1 = 0}, then ∨ β n -β 0 2 = O P p 1/2 0 ν n . Thus, the sequence (a n ) is in this case a n = p 1/2 0 ν n (see Theo- rem 1 of
Another possibility of estimator ∨ β n in (5) could be the expectile estimator calculated by (3). In this case, we study the convergence rate of

β n when p = O(n c ), 1/2 ≤ c ≤ 1, p ≤ n.
Then, in a similar way to Theorem 2.1(i), we have the following Lemma.

Lemma 3.1 Under assumptions (A1), (A2) and (A4), we have that β n -β 0 1 = O P (a n ), with the sequence (a n ) n∈N such that a n → 0 and n 1/2 a n → ∞.

By this lemma we deduce that another estimator ∨ β n,j in (5) can be the expectile estimator β n,j , for j = 1, • • • , n. We will compare in the next section, by simulations, these two possible choices of estimators in (5). Still in the next section, we will compare adaptive weights (5) with (4). The form of the random process R n (β) defined by (2) and the adaptive LASSO expectile estimator β n given by (3) remain the same, only the adaptive weight ω n,j can change, using either (5) or (4). It would be desirable for β n to satisfy the oracle proprties. For the sparsity property of β n its convergence in L 1 norm is required. In the following theorem, (b n ) n∈N is a deterministic sequence converging to 0 as n → ∞. This theorem holds for the two possible choices of ω n,j .

Theorem 3.1 Under assumptions (A1), (A2) and (A4), the tuning parameter (λ n ) n∈N and sequence

(b n ) satisfying λ n p 1/2 0 b -1 n → 0, as n → ∞, we have, β n -β 0 1 = O P (b n ).
The result of Theorem 3.1 indicates that the convergence rate of the adaptive LASSO expectile estimator β n depends on the chosen sequence (λ n ) n∈N . On the other hand, the convergence rate of β n doesn't depend on the convergence rate (a n ) of the estimators ∨ β n or β n . The only thing that matters (see relation ( 23) of the proof in Section 5) is that ∨ β n converges in probability to β 0 . The result of Theorem 3.1 now allows us to state oracle properties. For a quantile model, [START_REF] Zheng | Adaptive penalized quantile regression for high dimensional data[END_REF] obtains that the convergence rate, in L 2 norm, of the adaptive LASSO quantile estimator is (p 0 /n) -1/2 and that it also satisfies the oracle properties. The sparsity of the adaptive LASSO quantile estimator will be shown in our Section 4 by simulations, where we obtain that compared to the adaptive LASSO expectile estimator, it would be necessary to have a larger number n of the observations when the model errors have an asymmetric distribution. From where, a supplementary interest in considering the expectile method instead of the quantile.

Theorem 3.2 Suppose that assumptions (A1), (A2) and (A4) hold, the tuning parameter

(λ n ) and sequence (b n ) satisfy λ n p 1/2 0 b -1 n → 0, λ n b -1 n min n 1/2 , a -γ n → ∞, as n → ∞. Then: (i) P A n = A → 1, for n → ∞. (ii) For any vector u of size p 0 such that u 1 = 1, we have n 1/2 (u t Υ -1 n,A u) -1/2 u t ( β n - β 0 ) A L -→ n→∞ N 0, Var [gτ (ε)] E 2 [hτ (ε)] .
Hence, for p = O(n c ), with c ∈ [1/2, 1], the variance of the normal limit distribution is the same as that obtained when p = o(n 1/2 ). As for the case c < 1/2, studied in Section 2, the convergence rate of the adaptive LASSO espectile estimator doesn't depend on the power γ in the adaptive weight. However, γ intervenes in the imposed conditions so that the oracle properties are satisfied. If τ = 0.5, that is for the adaptive LASSO LS-estimator, we obtained the same asymptotic normal distribution as that given by Huang et al. ( 2008)) for their adaptive LASSO LS-estimator.

Regarding the tuning parameter sequence we make the following remark, useful for simulations and applications on real data.

Remark 3.1 The supposition λ n p 1/2 0 b -1 n → 0 made in the Theorems 3.1 and 3.2, implies that the tuning sequence λ n → 0, as n → ∞.

Remark 3.2 As for the non-penalized expectile estimator, by Theorems 3.1 and 3.2, we obtained that the asymptotic variance of the adaptive LASSO expectile estimators can be calculated without going via the density of the model errors. Contrarily, the variance of the adaptive LASSO quantile estimator depends on the density function of the error (see [START_REF] Ciuperca | Adaptive LASSO model selection in a multiphase quantile regression[END_REF]). More precisely, the variance of normal distribution limit of the adaptive LASSO expectile estimators can be estimated without estimation of the density function of the error ε.

Numerical study

In this section we first perform a numerical simulation study to illustrate our theoretical results on the adaptive LASSO expectile estimation and to compare it with the estimations obtained by the adaptive LASSO quantile method. Afterwards, an application on real data is presented. We use the following R language packages: package SALES with function ernet for the expectile regression and the package quantreg with function rq for quantile regression.

Given assumption (A1), the index τ is:

τ = E ε1 1 ε<0 E ε(1 1 ε<0 -1 1 ε>0 ) . ( 6
)
In the simulation study, the index τ is fixed, but it depends of the law of the model errors ε, such that assumption (A1) is satisfied.

Taking into account the suppositions imposed on the tuning parameter in Theorems 2.2 and 3.2, we consider λ n = n -2/5 for the expectile framework. For the quantile method, the tuning parameter is n 2/5 and the weight in the penalty have the power 1.225 (see [START_REF] Ciuperca | Adaptive LASSO model selection in a multiphase quantile regression[END_REF]).

In order to better compare the two methods, when the penalty weight for the adaptive LASSO expectile method is of type (4), then for adaptive LASSO quantile method is also of type (4). Similarly, if the adaptive weight is of type (5), it will be for both frameworks.

In the weights (4) or (5) of the expectile penalty, the used estimations are obtained, respectively, by expectile or LASSO expectile method. For the adaptive LASSO quantile method, the estimation in the weights are obtained, respectively, by quantile or LASSO quantile method.

4.1. Simulation study: fixed p 0 case In this subsection, we will study the numerical behavior of the adaptive LASSO expectile method and we will compare it with the simulation results obtained by the adaptive LASSO quantile method. For model (1), we consider p 0 = 6 and A = {1, • • • , 6}. In the all simulations of this subsection we take, β 0 1 = 1, β 0 2 = 4, β 0 3 = -3, β 0 4 = 5, β 0 5 = 6, β 0 6 = -1, while n and p are varied. For the errors ε i , three distributions are considered: N (0, 1) which is symmetrical, Exp(-1) and N (-1.2, 0.4 2 ) + χ 2 (1), the last two being asymmetrical. In Figure 1 we give the histogram for 10 5 realizations of a N (-1.2, 0.4 2 )+χ 2 (1) random variable, with empirical mean equal to -0.2 and median -0.628. The exponential law Exp(-1) has the density function exp(-(x + 1))1 1 x>-1 . For each value of n, p and distribution of ε, 1000 Monte Carlo replications are realized for two possible values for γ. In Tables 1 and2 we give the average Table 1: Sparsity study for expectile method with adaptive LASSO penalty, with weights of type (5), ε ∼ N (0, 1), two values for γ. Perfect method, if: Card(A ∩ An) = p0 = 6 and Card( An \ A) = 0. 5) are based on the respective LASSO estimations. For a perfect method, we should have:

n p Card(A ∩ An) Card( An \ A) γ = 1/8 γ = 2 γ = 1/8 γ =
Card(A ∩ A n ) = p 0 = 6 and Card( A n \ A) = 0.
In Table 1, for standard Gaussian errors, the values considered for γ are 1/8 and 2. Since for γ = 1/8 the number of false non-zeros, which in addition increases with n, is much larger than for γ = 2 and for γ = 2 the number of the true non-zeros decreases with n, these values will be dropped, other two values will be considered in Table 2. These results will be consolidated by the study on sparsity, for the three distributions of ε and different values for n and p. In Table 2, taking γ ∈ {5/8, 1}, for the adaptive LASSO expectile method with weights (5), all significant variables are detected when n is large enough, the number p of variables not coming into play. On the other hand, the smaller n, the worse the results obtained by the adaptive LASSO quantile method. In Table 3, in the adaptive weights of the expectile or quantile frameworks, we consider the expectile and quantile estimators, respectively. Comparing Tables 2 and3 we deduce that by the quantile framework for the three error distributions and by the expectile framework for exponential and mixing error distributions, the best results are for weights of type (4) when p n. Conversely, when p n, the performance is worse by using these weights for the two frameworks, the number of null parameters estimated as non null is large by the expectile method, specially for asymmetric errors. This conclusion is consolidated by Figures 2345678. When p = n, the R function rq cannot calculate the quantile estimations, since the matrix design is singular, from where the symbol "NA" (not available) in Table 3. Thus, the results obtained in Tables 2 and 3 show the advantage of using the adaptive LASSO expectile method compared to the adaptive LASSO quantile method.

We observe that the penalized expectile method better detects non-zero parameters compared to the penalized quantile method, especially for asymmetric errors. Concerning the two values considered for γ, when γ = 5/8, there are a little more true non-zeros detected, while, when γ = 1, there are fewer false non-zeros. This trend will be also confirmed by the following numerical studies.

Table 2: Sparsity study for expectile (ES) and quantile (Q) methods, with the adaptive LASSO weights of type ( 5). |A| = p0 = 6 and 1000 Monte Carlo replications. Perfect method, if: Card(A ∩ An) = p0 = 6 and Card( An \ A) = 0. In this subsection, we always compare expectile and quantile penalized methods, but when the values considered for p vary with n. Moreover, the number of non-zero parameters can increase as n increases. In Table 4 we take p = [n/2], p 0 = 2[n 1/2 ], with [x] the entire part of x, the power γ ∈ {5/8, 1} and ε ∼ Exp(-1). The true value of the non-null parameter vector is

ε n p Card(A ∩ An) Card( An \ A) ES Q ES Q γ = 5/8 γ = 1 γ = 5/8 γ = 1 N (-1.2, 0.4 2 ) + χ 2 (
Card(A ∩ An) = p0 = 6 and Card( An \ A) = 0. ε n p Card(A ∩ An) Card( An \ A) ES Q ES Q γ = 5/8 γ = 1 γ = 5/8 γ = 1 N (-1.2, 0.4 2 ) + χ 2 (
ε n 100p -1 0 Card(A ∩ An) 100(n -p 0 ) -1 Card( An \ A) mean(| β n -β 0 |) mean(|( β n -β 0 A |) ES Q ES Q ES Q ES Q γ = 5/8 γ = 1 γ = 5/8 γ = 1 γ = 5/8 γ = 1 γ = 5/8 γ = 1 N (0,
β 0 A = (1, • • • , p 0 ).
: p = [n(log n) -1 ], p0 = 2[n 1/4 ], ε ∼ Exp(-1).
The adaptive weights are of type (5).

β 0 A n 100p -1 0 Card(A ∩ An) 100(n -p 0 ) -1 Card( An \ A) mean(| β n -β 0 |) mean(|( β n -β 0 A |) ES Q ES Q ES Q ES Q γ = 5/8 γ = 1 γ = 5/8 γ = 1 γ = 5/8 γ = 1 γ = 5/8 γ = 1 (1, 2
β 0 |) = (M p) -1 M m=1 p j=1 | β (m) n,j -β 0 j |. Similarly we calculate mean(|( β n -β 0 ) A |) = (M p) -1 M m=1 p 0 j=1 | β (m) n,j -β 0 j |.
The results are similar by the two estimation methods, while by the adaptive LASSO expectile method the results being more accurate for γ = 1 than for γ = 5/8. In Table 5, taking p = [n(log n) -1 ], the value of p is increased compared to that considered in Table 4. Furthermore, the sparsity of the model is more accentuated by considering

p 0 = 2[n 1/4 ]. Two values for β 0 A are considered: (1, 2, • • • , p 0 ) and 1 p 0 = (1, • • • , 1
) while for the model errors, only the exponential distribution ε ∼ Exp(-1) is made. For both values of β 0 , the expectile and quantile methods with adaptive LASSO penalty give very good results for identifying of null and non-null parameters. Comparing Tables 4 and5, for n fixed and exponential law, we deduce that the penalized expectile estimation quality of the model does not vary for two different p. Furthermore, the quality is better if p 0 decreases and when γ = 1. We also deduce that the quality of automatic detection increases with n.

In Table 6 we give the median run time, for 1000 simulation replicates, of R functions: "ernet" for calculate the adaptive LASSO expectile estimation and "rq" for adaptive LASSO quantile estimation, for a linear model with |A| = p 0 = 6, ε ∼ Exp(-1), γ = 1 and adaptive LASSO weights of type (4). The algorithm used for "ernet" has been proposed by [START_REF] Gu | High-dimensional generalizations of asymmetric least squares regression and their applications[END_REF] and combines the cyclic coordinate descent and proximal gradient algorithms. The calculation time of the adaptive LASSO quantile estimate is much longer than that of the penalized expectile method, especially when the number p of parameters is close to the sample size n.

Sparsity study function of γ

In Figure 2 we present the results of the percentages of the true non-zero and false zero estimators by adaptive LASSO expectile method, for a model with n = p = 100 and Gaussian estimators. The true parameters are β 0 1 = 1, β 0 2 = 4, β 0 3 = -3, β 0 4 = 5, β 0 5 = 6, β 0 6 = -1 and β 0 j = 0 for any j > 6. The results are better when in the adaptive weights

Table 6: Run time (in seconds) of R functions: "ernet" corresponding to the expectile method and "rq" for quantile method, both being LASSO penalized with LASSO penalty, for a model: |A| = p0 = 6, ε ∼ Exp(-1), γ = 1, adaptive LASSO weights of type (4).

n p time by adaptive LASSO expectile (by "ernet") time by adaptive LASSO quantile (by "rq") 50 10 9.9 • 10 -4 9.9 • 10 -4 25 9.9 • 10 -4 9.9 • 10 -4 200 10 9.9 • 10 -4 10 -3 100 2 • 10 -3 10 -2 500 10 9.9 • 10 -4 1.9 • 10 -3 250 2.9 • 10 -3 1.1 • 10 -1 1000 10 10 -3 2.99 • 10 -3 500 10 -2 0.93 10 4 10 4.9 • 10 -3 0.03 10 3 0.29 38.9

we take the LASSO expectile estimator. Following this results, right now, for weight (5) we consider only LASSO expectile estimator.

In Figures 345678, for n = 100, p ∈ {10, 100}, ε ∼ N (0, 1) or Exp(-1) or N (-1.2, 0.4 2 )+χ 2 (1), we present the percentage of true and false zero, when for the adaptive weight we take (5) by dotted line, and (4) with solid line. We deduce that for p n we must take weights (4), for p n weights (5), choosing in the two cases γ ∈ [1/2, 1]. In Figures 9 and 10, for ε ∼ N (-1.2, 0.4 2 ) + χ 2 (1), for the same n (either 50 or 100), we vary p. Then, when p = n we consider adaptive weight (5) with dotted line and in the other case we take (4) with solid line. We deduce that we must take different weights, function that p is of the same order as n or much smaller.

Conclusion of simulation study

For adaptive LASSO expectile estimation, when p n, the best results are obtained for adapted weights (4) and when p n for weights (5). The penalized expectile method is more accurate when n is small and the error distributions are asymmetric. When p and p 0 increase with n, the adaptive LASSO expectile estimations identify very well the null and non-zero parameters, while the adaptive LASSO quantile estimations happen to have the same performances only for very large n. The calculation time of the adaptive LASSO quantile estimate is much longer than that of the adaptive LASSO expectile method, especially when the number p of parameters is close to n. On the other hand, the adaptive LASSO quantile method identifies true zeros less well when the number of observations is small. Moreover, another disadvantage of the penalized quantile method is that when the number of parameters is close to the number of observations, the R function cannot calculate numerically the quantile estimations. In all cases, the most appropriate the power γ which intervenes in adaptive weights of the penalized expectile method is γ ∈ [1/2, 1].

Application to real data

We use the data eyedata of R package flare which contains n = 120 observations (rats) for the response variable of gene TRIM32 and 200 explanatory variables, other genes probes, from the microarray experiments of mammalian-eye tissue samples in [START_REF] Scheetz | Regulation of gene expression in the mammalian eye and its relevance to eye disease[END_REF]. The objective is to find genes that are correlated with the TRIM32 gene, known to cause Bardet-Biedl syndrome, a genetically disease of multiple organ systems including the retina. In this paper, the expectile index τ is supposed to be known, such that for model error ε we have E[g τ (ε)] = 0. On the other hand, in applications, an estimate should be given for τ , with the remark that same related question also exists in quantile regression for estimating the quantile level τ such that IP [ε < 0] = τ . But to estimate τ we must first calculate the model errors and to calculate the errors we must know τ . Therefore, we will not give an exact estimator of τ but we will try to prefix the value of τ such that the dispersion around their median of the values of (y i ) 1 i n is taken into account. We propose the following approach in order to give a prefixed value for the expectile index τ . The values of the explained variable are first transformed, for i

= 1, • • • , n, y i = (y i -M n )/D n , with M n = median(y 1 , • • • , y n ) and D n = n -1 n i=1 |y i -M n |
the empirical mean of the absolute values of the distances to the median of the y i . Afterwards, based on relation (6), we calculate the empirical estimation of τ for y i :

τ = n -1 n i=1 y i 1 1 y i <0 n -1 n i=1 y i 1 1 y i <0 -n i=1 y i 1 1 y i >0
.

Then, in model ( 1), the response variable is

Y i = (Y i -M n )/D n .
For this application, we get τ = 0.568 and γ = 5/8. We first consider two linear models (the first on the first hundred explanatory variables and the second on the other hundred explanatory variables), for which we consider adaptive weights of type ( 5), for γ = 5/8, λ n = n 2/5 . Afterwards we consider a linear model with explanatory variables the relevant regressors selected for the two preceding regressions. For this model, we select the relevant variables by the adaptive LASSO expectile method with the weights of type (4). Then, we obtain that the genes whose expressions influence gene TRIM32 are 87,153,180,185,200 with the labels: "21092", "25141", "28680", "28967" and "30141". The obtained estimations for the coefficients of these four explanatory variables are respectively: -1.06, 2.97, 1.29, -1.727 and -0.23. In Figure 11 we illustrate the histogram and the box-plot for response variable Y i . We observe that there are outliers. If a classical LS regression of the TRIM32 variable in respect to the five selected covariates is performed, we obtain a model with an adjusted R 2 = 0.75. The all five variables are significant and the residuals have Gaussian distribution (the p-value by Shapiro test equal to 0.69). In the sub-figure of the right-hand side of Figure 11, we also present, the forecasts beside of the true values of TRIM32. We observe that the scater graph is on the first bisectrix. By the adaptive LASSO quantile method, no variable is selected among the 200 explanatory ones.

In literature works that model the same data, variable number 153, tagged "25141", has been selected as the sole regressor by the bayesian shrinkage in [START_REF] Song | Nearly optimal Bayesian Shrinkage for High Dimensional Regression[END_REF] and by a globally adaptive quantile method in [START_REF] Zheng | Globally adaptive quantile regression with ultra-high dimensional data[END_REF] for quantile index between 0.45 and 0.55. In this last paper, there are other covariates that appear to be significant for other quantile index values. These variables are: "11711", "24565", "25141", "25367", "21092", "29045", "25439", "22140", "15863" and "6222". If we make a classic regression for these ten regressors, we obtain, with a risk of 0.05 that only the variables "25141", "21092", "29045", "15863", "6222" are significant, in a model of lower quality (adjusted R 2 = 0.72, residual standard error=0.52) than the one with the five explanatory variables found in the present paper.

In order to complete the comparison of the three methods, we split the database of 120 observations into two: one for learning and the other for testing. The three methods 2015), while by the adaptive LASSO quantile method it is less good, also because of the fact that no explanatory variable is significant for the latter. From this table, we also deduce that the method proposed in the present paper is also robust because we get the same accuracy on the test and learning set.

Proofs

In this section we give the proofs of the results presented in Sections 2 and 3.

Result proofs in Section 2

Before presenting the proof of Theorem 2.1, let's recall a result given in [START_REF] Gu | High-dimensional generalizations of asymmetric least squares regression and their applications[END_REF].

Lemma 1 of [START_REF] Gu | High-dimensional generalizations of asymmetric least squares regression and their applications[END_REF]. For any z, z 0 ∈ R and τ ∈ (0, 1) we have:

min(τ, 1 -τ )(z -z 0 ) 2 ≤ ρ τ (z) -ρ τ (z 0 ) -g τ (z 0 )(z -z 0 ) ≤ max(τ, 1 -τ )(z -z 0 ) 2 .
Proof of Theorem 2.1. (i) In order to show the convergence rate of the expectile estimator, we show that for all > 0, there exists B > 0, large enough when n is large, such that:

P inf u∈R p , u 2 =1 Q n β 0 + B p n u > Q n (β 0 ) ≥ 1 -. (7)
This part of proof is similar to that of Lemma 1.2 in Zhao et al. (2018), for the convergence rate of the oracle estimator. Let B > 0 be a constant to be determined later and u a vector in R p with the norm u 2 = 1. Let's study the difference:

Q n β 0 + B p n u -Q n (β 0 ) = n i=1 ρ τ ε i -B p n X t i u -ρ τ (ε i ) = n i=1 ρ τ ε i -B p n X t i u -ρ τ (ε i ) -E ρ τ ε i -B p n X t i u -ρ τ (ε i ) + n i=1 E ρ τ ε i -B p n X t i u -ρ τ (ε i ) ≡ ∆ 1 + ∆ 2 .
We first study the term ∆ 2 . By Taylor expansion, we have:

E[ρ τ (ε -t) -ρ τ (ε)] = E -g τ (ε)t + 2 -1 h τ (ε)t 2 + o(t 2 ) = 2 -1 E h τ (ε) t 2 + o(t 2 )
. By the Cauchy-Schwarz inequality, we have that |X t i u| 2 ≤ X i 2 2 u 2 2 and then, using assumption (A3), we obtain that (pn -1 ) 1/2 max 1 i n |X t i u| -→ n→∞ 0. Thus,

∆ 2 = 1 2 n i=1 B 2 p n X t i u 2 E h τ (ε) + o B 2 p n X t i u 2 E h τ (ε) .
On the other hand,

2 min(τ, 1 -τ ) ≤ E h τ (ε) = 2τ E[1 1 ε≥0 ] + 2(1 -τ )E[1 1 ε<0 ] ≤ 2 max(τ, 1 -τ ). Then 0 < ∆ 2 = B 2 p n E h τ (ε) n i=1 X t i u 2 1 + o(1) = O(B 2 p). (8)
We are now studying the term ∆ 1 . Let us consider the random variable:

D i ≡ ρ τ ε i -B p n X t i u -ρ τ (ε i ) + B p n g τ (ε i )X t i u. (9)
Then, we can write ∆ 1 as:

∆ 1 = n i=1 -B p n g τ (ε i )X t i u + D i -E[D i ] .
Using assumption (A1) we have, E (pn -1 ) 1/2 g τ (ε i )X t i u = 0 and Var (pn

-1 ) 1/2 n i=1 g τ (ε i )X t i u = pn -1 u t n i=1 X i X t i uVar [g τ (ε)] = O(p)
. Then, we have:

B p n n i=1 g τ (ε i )X t i u = O P p 1/2 B . (10)
By Lemma 1 of [START_REF] Gu | High-dimensional generalizations of asymmetric least squares regression and their applications[END_REF] for z = ε i -B(pn -1 ) 1/2 X t i u and z 0 = ε i , we can write D i also in the form: D i = B 2 pn -1 |X t i u| 2 V i , with V i a random variable between min(τ, 1-τ ) and max(τ, 1 -τ ) with probability one:

P min(τ, 1 -τ ) ≤ V i ≤ max(τ, 1 -τ ) = 1. (11) Then, Var [D i ] ≤ E[D 2 i ] = B 4 (p/n) 1/2 |X t i u| 4 E V 2 i . But E V 2 i ≤ 1 and thus Var [D i ] ≤ B 4 p 2 n -2 |X t i u| 4 .
On the other hand, the random variables D i defined by ( 9), are independent. Then,

n i=1 D i -E[D i ] = O P n i=1 Var [D i ] ≤ O P n i=1 E[D 2 i ] = O P B 2 p √ n . (12)
Relations ( 10) and ( 12), imply, since pn -1 → 0, when n → ∞, that:

∆ 1 = O P (Bp 1/2 ) + O P (B 2 pn -1/2 ) = O P (Bp 1/2 ).

Then, this last relation together relation (8) imply ∆ 2 > |∆ 1 |, with probability converging to one, for B large enough. Relation (7) follows, which implies the convergence rate of the expectile estimator.

(ii) For p-vector u = (u 1 , • • • , u p ), with u 2 = 1 and B > 0 a constant, let us consider the difference

R n β 0 +B p n u -R n (β 0 ) = Q n β 0 +B p n u -Q n (β 0 )+nλ n p j=1 ω n,j β 0 j +B p n u j -|β 0 j | . ( 13
)
The first term of the right-hand side of (13) becomes by the above proof for (i),

Q n β 0 + B p n u -Q n (β 0 ) = B 2 p n E h τ (ε)
Since p 1/2 0 n (1-c)/2 λ n → 0, as n → ∞, we obtain that relation ( 14) dominates ( 15) and the assertion regarding the convergence rate of β n results.

Proof of Theorem 2.2. (i) Let us consider the parameter set: V p (β 0 ) ≡ β ∈ R p ; ββ 0 2 ≤ B p n , with B > 0 large enough and W n ≡ β ∈ V p (β 0 ); β A c 2 > 0 . According to Theorem 2.1, the estimator β n belongs to the set V p (β 0 ) with a probability converging to 1 as n → ∞. In order to show the sparsity property of claim (i), we will show that, lim n→∞ P β n ∈ W n = 0. Note that if β ∈ W n , then p > p 0 . Let us consider two parameter vectors: β = (β A , β A c ) ∈ W n and β (1) = (β (1)

A , β (1)
A c ) ∈ V p (β 0 ), such that β A c = 0 p-p 0 . For this parameters, we will study the following difference:

n -1 R n (β) -R n (β (1) ) = n -1 Q n (β) -Q n (β (1) ) + λ n p j=p 0 +1
ω n,j |β j |.

(16)

First, note that by elementary calculations, we get: ρ τ (ε -t) = ρ τ (ε) -g τ (ε)t + 1 2 h τ (ε)t 2 + o P (t 2 ) for t → 0. Then, for the first term of the right-hand side of relation ( 16), we have

n -1 n i=1 ρ τ (Y i -X t i β (1) ) -ρ τ (Y i -X t i β) = n -1 n i=1 ρ τ ε i -X t i,A (β A -β 0 A ) -ρ τ ε i -X t i,A (β A -β 0 A ) -X t i,A c β A c = n -1 n i=1 g(ε i )X t i,A β A -β 0 A + h(ε i ) 2 X t i,A (β A -β 0 A ) 2 + o P X t i,A (β A -β 0 A ) 2 -n -1 n i=1 g(ε i )X t i β A -β 0 A , β A c + h(ε i ) 2 X t i (β A -β 0 A , β A c ) 2 + o P X t i (β A -β 0 A , β A c ) 2 .
By similar arguments used in the proof of Theorem 2.1(i) we have

n -1 n i=1 g(ε i )X t i,A β A -β 0 A = E n -1 n i=1 g(ε i )X t i,A β A -β 0 A + O P Var n -1 n i=1 g(ε i )X t i,A β A -β 0 A 1/2 = O P Var n -1 n i=1 g(ε i )X t i,A β A -β 0 A 1/2 ≤ O P E[g 2 (ε i )] 1 n 2 n i=1 X i,A 2 2 β A -β 0 A 2 2 1/2 = O P 1 n p n 1/2 = O P p 1/2 n .
Proceeding similarly as above, we get:

n -1 n i=1 g(ε i )X t i β A -β 0 A , β A c = O P p 1/2 n .
Taking into account relation ( 11), we deduce that: 0

< n -1 n i=1 h(ε i ) X t i (β A -β 0 A , β A c ) 2 = O P ( β A -β 0 A 2
2 ) = O P (pn -1 ) and also that,

0 < n -1 n i=1 h(ε i ) X t i,A (β A -β 0 A ) 2 = O P p n .
By these relations, we obtain that the first term of the right-hand side of relation ( 16) is of order pn -1 . For the penalty of the right-hand side of relation ( 16), taking into account Theorem 2.1(i) and since β ∈ W n we obtain:

λ n p j=p 0 +1
ω n,j |β j | ≥ Cλ n p n

(1-γ)/2

.

Using the supposition λ n (pn -1 ) -(1+γ)/2 -→ n→∞ ∞, that is λ n n (1-c)(1+γ)/2 -→ n→∞ ∞, we have that in the right-hand side of relation ( 16), it's the penalty that dominates. Then, since n -1 Q n (β) -Q n (β (1) ) = O P (pn -1 ), we have,

n -1 R n (β) -R n (β (1) ) ≥ Cλ n p n (1-γ)/2 . ( 17
)
But, on the other hand, since β

(1)

A c = 0 p-p 0 , by similar arguments as above, we have, n -1 R n (β 0 ) -R n (β (1) ) = O P (pn -1 ). From the last relation together relation (17), since λ n (pn -1 ) -(1+γ)/2 -→ n→∞ ∞, we deduce, lim n→∞ P[ β n ∈ W n ] = 0.

Figure 1 :

 1 Figure 1: Histogram for 10 5 realizations of a N (-1.2, 0.4 2 ) + χ 2 (1) random variable.

  We assess model selection by calculating the percentage (100p -1 0 Card(A ∩ A n )) of the non-zero parameters with a non-zero estimation and the percentage of false significant variables (100(n-p 0 ) -1 Card( A n \A)), by the two estimation methods. We also give the accuracy of the complete estimation vectors (mean(| β n -β 0 |)) and of the estimations of non-zero parameters (mean(|( β n -β 0 A |)) (average absolute estimation error) obtained on 1000 Monte Carlo replications. More precisely, if M is the Monte Carlo replication number and β (m) n,j is the estimation of β 0 j obtained for the Monte Carlo

Figure 2 :

 2 Figure 2: Percentage of true and false non zero, for ε ∼ N (0, 1), n = 100 and p = 100. In adaptive weights (5), two estimators: LASSO expectile with dotted line and expectile with solid line.

Figure 3 :

 3 Figure 3: Percentage of true and false non zero, for ε ∼ N (0, 1), n = 100 and p = 10. Two adaptive weights: (5) with dotted line and (4) with solid line.

Figure 4 :

 4 Figure 4: Percentage of true and false non zero, for ε ∼ N (0, 1), n = 100 and p = 100. Two adaptive weights: (5) with dotted line and (4) with solid line.

Figure 5 :

 5 Figure 5: Percentage of true and false non zero, for ε ∼ Exp(-1), n = 100 and p = 10. Two adaptive weights: (5) with dotted line and (4) with solid line.

Figure 6 :

 6 Figure 6: Percentage of true and false non zero, for ε ∼ Exp(-1), n = 100 and p = 100. Two adaptive weights: (5) with dotted line and (4) with solid line.

Figure 7 :

 7 Figure 7: Percentage of true and false non zero, for ε ∼ N (-1.2, 0.4 2 ) + χ 2 (1), n = 100 and p = 10. Two adaptive weights: (5) with dotted line and (4) with solid line.

Figure 8 :

 8 Figure 8: Percentage of true and false non zero, for ε ∼ N (-1.2, 0.4 2 ) + χ 2 (1), n = 100 and p = 100. Two adaptive weights: (5) with dotted line and (4) with solid line.

Figure 9 :

 9 Figure 9: Percentage of true and false non zero, for ε ∼ N (-1.2, 0.4 2 ) + χ 2 (1), n = 50. Two value for the number of parameters: p = 10 together adaptive weight (5), with dotted line and p = 50 together adaptive weight (4), with solid line.

Figure 10 :

 10 Figure 10: Percentage of true and false non zero, for ε ∼ N (-1.2, 0.4 2 ) + χ 2 (1), n = 100. Two value for the number of parameters: p = 50 together adaptive weight (5), represented with dotted line and p = 100 together adaptive weight (4), represented with solid line.

Figure 11 :

 11 Figure 11: Histogram and boxplot of ( yi) 1 i 120 . Scater graph between forecast and the true value of TRIM32.

  2 1 + o P (1) = O P (B 2 p). (14)Furthermore, for the penalty of (13) we have:

  of the 1000 Monte Carlo replications for the cardinalities (number of the true non-zeros estimated as non-zero) Card(A ∩ A n ) and Card( A n \ A) (number of the false non-zero) by the expectile (ES) and quantile (Q) penalized methods, each with LASSO adaptive penalty. The weights of type (

						2
	50	10	5.99	5.74	0.22	0
		25	5.99	5.7	1	0.001
		50	5.99	5.64	2.13	0.002
	100	10	6	5.99	0.13	0
		25	6	5.99	0.56	0
		100	6	5.98	2.5	0

Table 3 :

 3 Sparsity study for expectile (ES) and quantile (Q) methods, with adaptive LASSO weights of type (4). |A| = p0 = 6 and 1000 Monte Carlo replications. Perfect method, if:

Table 5 :

 5 Study of the sparsity evolution and of the estimation accuracy for the expectile (ES) and quantile framework, with p and p0 depending an n

Table 7 :

 7 Means of residual absolute values (RAV) by adaptive LASSO expectile estimation, LS on explanatory variables of[START_REF] Zheng | Globally adaptive quantile regression with ultra-high dimensional data[END_REF], and by adaptive LASSO quantile estimation on all data, learning and test data. each calibrated on 105 observations and are tested on 15 observations. The empirical means of residual absolute values (RAV) are presented in Table7. The forecast accuracy of the explained variable is very similar by the adaptive LASSO expectile and by the LS estimation on explanatory variables ofZheng et al. (

	estimation method	RAV on all data	RAV on learning data	RAV on test data
	adaptive LASSO expectile	0.057	0.056	0.068
	Zheng et al. (2015) method	0.059	0.059	0.064
	adaptive LASSO quantile	0.75	0.787	0.771

are

(ii) Given the previous result we consider the parameter vector β of the form: β = β 0 + (pn -1 ) 1/2 δ, with δ = (δ A , δ A c ), δ A c = 0 p-p 0 , δ A 2 ≤ C. We study then the following difference:

For the penalty P = λ n p 0 j=1 ω n,j |β j |-|β 0 j | of the right-hand side of relation ( 18) we have, by 2.1(i), ω n,j = | β n,j | -γ = O P (1) and by the triangular inequality |β j | -|β 0 j | ≤ |β j -β 0 j |. Then, as in the proof of Theorem 2.1, by relation (15), we obtain:

For the first term of the right-hand side of relation ( 18) we have:

which has as minimizer the solution of

from where, we get,

We deduce that, the minimum value of (20) is of order O P pn -1 δ A 2 = O P pn -1 = O P n c-1 . Taking into account the supposition λ n p 1/2 0 n (1-c)/2 -→ n→∞ 0 and relation ( 19), we have that, P = o P (pn -1 ). Then, in the right-hand side of relation ( 18), the first term is the dominant one.

Let us now consider the following random variable sequence:

Thus, taking into account assumption (A1), we get:

which implies

The proof of claim (ii) is finished.

Result proofs in Section 3

Proof of Lemma 3.1.

The proof is similar to that of Theorem 2.1(i). Consequently, we give only the main results. Let us consider the p-vector u = (u 1 , • • • , u p ), with u 1 = 1. By Holder's inequality: |X t i u| ≤ X i ∞ u 1 , using also assumption (A4) we obtain that max 1 i n |X t i u| < ∞. For a constant B > 0, we will study the difference:

where, since a n → 0, n 1/2 a n → ∞, we get that ∆ 2 > ∆ 1 with probability converging to one, for B large enough.

Proof of Theorem 3.1. The proof is similar to that of Theorem 2.1(ii). Consequently, we give only the main results. Otherwise, instead of the Cauchy-Schwarz inequality we use Holder's inequality:

). For a p-vector u = (u 1 , • • • , u p ), with u 1 = 1 and a constant B > 0, let the difference

By a similar approach made for the terms ∆ 1 and ∆ 2 of the proof of Theorem 2.1, we obtain:

For the penalty of the right-hand side of relation ( 21) we have:

by the Cauchy-Schwarz inequality and afterwards by the estimator consistency of

Since λ n p 1/2 0 b -1 n → 0, as n → ∞, then relation ( 22) dominates ( 23) and the theorem follows.

Proof of Theorem 3.2. (i) Let j ∈ A c be, then j > p 0 . Thus, the derivative of the random process R n (β) in respect to β j is:

For the first term of the right-hand side of relation ( 24), we have,

. By elementary calculations, we can show that, for t → 0, we have:

On the other hand, by the Holder's inequality, we have:

Using assumption (A4) and Theorem 3.1, we obtain that

By the Central Limit Theorem, tacking into account assumption (A4), we have that:

). On the other hand, 0 < h τ (ε i ) < 2 with probability 1. Using the Holder's inequality, we have, with probability one, n i=1

from where, tacking into account assumption (A4), we have:

For the penalty of relation ( 24) we have:

as n → ∞, also taking into account relation ( 25) we have that:

Then, the solution of ( 24) must be equal to 0. From where β n,A c = 0 p-p 0 , with probability converging to 1. This relation implies A n ⊆ A with probability converging to 1 when n → ∞.

On the basis of this result, from now on we consider the parameters β of the form β = β A , 0 p-p 0 . We must show now that A ⊆ A n . By Theorem 3.1 we have β A -β 0 A 1 = O P b n , from where for any j = 1, • • • , p 0 , we obtain, β n,j P -→ n→∞ β 0 j = 0. Thus, since b n -→ n→∞ 0, we have that β n,j = 0 with probability converging to 1, from where A ⊆ A n .

(ii) Given the previous result (i) and Theorem 3.1, we consider the parameters β of the form: β = β 0 + b n δ, with δ = (δ A , δ A c ), δ A c = 0 p-p 0 , δ A 1 ≤ C. For the penalty P of the right-hand side of relation ( 18 . For the main part, we have:

The end of the proof is similar to that of Theorem 3.1(ii).


Digital Startup Access to Venture Capital Financing: What Signals Quality?

Keywords: Startups, Entrepreneurship, Financing, Digital Signals, Human capital, Social media networking, Venture capital, Signals

What signals quality for digital startups seeking access to external finance? Analyzing a unique dataset from India, we investigate the impact of traditional quality signals (e.g., the founders' years of experience, prior industry experience); networking signals (e.g., degrees from elite educational institutes, the breadth of an entrepreneurial team); and digital signals (e.g., the social media presence of the startup on multiple social media sites) on the access to venture capital financing. We find that while networking and digital signals positively impact access to financing, traditional human capital signals do not have any significant impact on the process.

INTRODUCTION

Entrepreneurial startups play a fundamental role in the promotion of innovation and economic growth (Acs & Audretsch, 1988;Acs et al., 2008;Audretsch, 2007a and2007b;Audretsch et al., 2009;Baumol, 1996;Chavis et al., 2011;Cumming & Johan, 2014;Cumming et al., 2014;Fairlie & Chatterji, 2013;Klapper & Love, 2011;Marcotte, 2012;McMullen, 2011;Naude, 2010;Stam & Wennberg, 2009;Thurik et al., 2008). The creation of new entrepreneurial and innovative ventures requires capital financing and, as such, there is a large body of research looking into access to entrepreneurial finance for entrepreneurs and startups (Bhide, 1992;Bonini et al., 2012;Fossen, 2014;Nahata, 2008;Wang & Wang, 2012;Li et al., 2016). We know that sustained financing can ensure high levels of growth and survival of new ventures (Cosh et al., 2009;Smallbone et al., 2003;Storey, 2016;Van Auken & Carter, 1989) while lack of financing can result in low levels of growth and an increased likelihood of failure (Alsos et al., 2006;Basu & Parker, 2001). As financing is crucial for startups not only at the initial stages but also at the later stages for continued growth and survival, extant research has sought to determine the factors that play a significant role in access to finance, such as the human capital of the founders (Almus & Nerlinger, 1999;Colombo & Grilli, 2005;Cressy, 1996;Cumming & Johan, 2008a;Stuart & Abetti, 1990;Westhead & Cowling, 1995) and founders' social networks (Goh et al., 2013;Hochberg et al., 2007;Luo et al., 2013) .

Human capital plays a significant role in entrepreneurial endeavors. Researchers have divided human capital into two broad categories: formal and informal. Formal signifies individual educational qualifications, skills, and abilities, whereas informal signifies human capital developed through work and familial experiences. We know that educational investments and work experience can translate into economic advantages (Becker, 1964;Killeen et al., 1999;Langelett, 2002) and, if applied towards an entrepreneurial activity, can result in improved entrepreneurial judgment (Baron & Ensley, 2006;Corbett, 2005;McGrath & MacMillan, 2000;Parker, 2006;Ronstadt, 1988;Shane, 2000;Wiklund & Shepherd, 2003) and forecasting ability (Clement, 1999;Mikhail et al., 1997). It is clear that the impact of human capital on entrepreneurial activity and judgment has been previously widely studied in more developed economies. There is however limited empirical evidence on the role of human capital in startup financing for firms in developing or emerging markets where institutions are arguably weaker, and informal quality signals are much more important to mitigate information asymmetry. In this paper we analyze Indian tech startups to determine the extent formal and informal human capital signals serve as positive signals to mitigate information asymmetries and potentially impact access to venture capital (VC) for Indian entrepreneurs. Our formal and informal measures include total years of experience, the type of work experience (technical, founder, or consultant), the degree to which an elite educational institute provides access to a network of resources, and the breadth of the entrepreneurial team.

In India, technology has been a major driver of innovation for the last 25 years. Major business opportunities have surfaced in the technological front; consequently, there has been exponential growth in the number of startups, which has attracted investments from a large number of venture capitalists (VCs).

From 2012-2017, reports show a significant increase in investments by VCs in Indian startups. The transaction value has increased over 13.5 times (from $94 million in 2011 to $1.275 billion in 2016) 1 . In 2017, investments by VCs in the country reached record levels at $26.5 billion.

With such rapid growth, policymakers face considerable challenges. One challenge is mitigating the inefficiency of the matching process between investors and new technological startups, especially in a financial market which still largely remains imperfect (e.g. Colombo & Grilli, 2005, 2009). In India, graduates from top engineering colleges are no longer restricted to only joining the corporate world; thus, those with engineering, mathematical, and analytical skills are increasingly incentivized to create their own entrepreneurial firms. A report published by NASSCOM 2 states that 50% of the founders of new firms have engineering backgrounds, 25% of them have management backgrounds, and 25% have other educational qualifications. In 2018, nearly half of the startups in India had one or more founders who studied in one or more of the country's top four engineering and management colleges: the IITs 3 , IIMs 4 , BITS 5 , or the Indian School of Business 6 . In the United States, we observe the same trend in relation to Harvard, Stanford, and MIT 7 according to Pitchbook 8 and Crunchbase statistics. These considerations raise our initial question of whether the traditional human capital signal of previous work experience continues to signal quality to investors, as suggested by extant research. Our follow-up question is whether a degree from an elite institute, which may signal quality and increased social capital due to a See the full report at: https://smartnet.niua.org/sites/default/files/resources/nasscom-start-up-report-2017.pdf.

3 Indian Institute of Technology, a premier engineering college.

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  PwC report 2016 2 NASSCOM stands for the National Association of Software and Service Companies, a nonprofit organization established in 1988. It initiated a program called 10000 Startups in 2013. The purpose is to support the impact of 10000 startups in India by 2023.