domain: math stat

text:
Keywords: Asymptotic normality, Bandwidth selection, Extreme value, Non parametric estimation, Mixing Data, Pareto distribution, Recursive estimator, Stochastic approximation algorithm, Strong consistency, Tail index. Chapter 2 Extreme value, Pareto distribution, Stochastic approximation algorithm, Tail index. for Estimation of a Distribution Function under α-mixing condition Asymptotic normality, Bandwidth selection, Mixing Data, Recursive distribution estimator, Stochastic approximation algorithm. Chapter 4 Density estimation, Mode, Non parametric estimation, Stochastic approximation, Strong consistency, Strong mixing

The main objective of this thesis resides in applying the stochastic approximation method to build up a large class of recursive non parametric kernel estimators for dependent and independent variables. First, we dene a recursive kernel estimator of the conditional extreme value index. We investigate the properties of the proposed recursive estimator and compare it to Hill's non recursive kernel estimator. We show that using some optimal parameters, the proposed recursive estimator dened by the stochastic approximation algorithm proves to be very competitive to Hill's estimator. Eciency and feasibility were conrmed by theoretical results and then by applications on simulated real data about Malaria in Senegalese children. Second, we extend the work of Slaoui (2014b) to the case of α-mixing data. We study the properties of these estimators and compare them with Nadaraya's non recursive distribution estimator. Using an optimal choice of the bandwidth and an appropriate choice of the stepsize parameter, the recursive estimators allowed us to obtain quite better results compared to the non recursive distribution estimator under α-mixing condition in terms of estimation error. We elaborate the central limit theorem and the uniform convergence for the proposed estimators under some mild conditions. The obtained theoretical results are corroborated through simulation study. Finally, we adopt the stochastic approximation algorithms to dene a kernel estimator of the mode based on the recursive kernel density estimator developed by Mokkadem et al. (2009a). Additionally, we establish its almost sure convergence under strong mixing hypothesis and we conrm these theoretical results through numerical simulations.

for all their support and friendship. Immense gratitude as always to Jihen, Habiba, Rym and Mohamed, then all my professors and classmates at ESSTHS for their encouragement.

I gratefully recognize the unconditional and endless love and support of my family. Mom, thank you for your uncountable prays for me. Dad, your sacrice is the most powerful source for me to accomplish my research and fulll your dream. I wish that you will be always proud of me. My brothers: Mustapha, Mokhles and Aymen, thank you for all the kind words and guidance you provided. Similarly other family members are also subjects to special thanks for their encouragement and faith on me during my studies. Last but not least, I am eternally grateful to my cousin Marwa for being patient and supportive through the dicult moments. She has always stood behind me, and been my source of inspiration.

Résumé

L'objectif de cette thèse réside dans l'application de la méthode d'approximation stochastique pour construire une classe d'estimateurs à noyau récursifs non paramétriques pour les variables dépendantes et indépendantes. Dans un premier temps, nous dénissons un estimateur récursif à noyau de l'indice conditionnel des valeurs extrêmes. Nous étudions les propriétés de l'estimateur récursif proposé et le comparons à l'estimateur à noyau non récursif de Hill. Nous montrons qu'en utilisant certains paramètres optimaux, l'estimateur récursif proposé déni par l'algorithme d'approximation stochastique s'avère très compétitif par rapport à l'estimateur de Hill. L'ecacité est conrmée par des résultats théoriques puis par des applications sur des données réelles simulées concernant le paludisme chez les enfants sénégalais. Deuxièmement, nous étendons le travail de [START_REF] Slaoui | The stochastic approximation method for the estimation of a distribution function[END_REF] au cas des données α-mélangeantes. Nous étudions les propriétés de ces estimateurs et les comparons avec l'estimateur de distribution non récursif de Nadaraya. En utilisant un choix optimal de la fenêtre et un choix approprié de pas, les estimateurs récursifs nous permettent d'obtenir de meilleurs résultats que l'estimateur de distribution non récursif dans le cas α-mélangeant en termes d'erreur d'estimation. Nous établissons le théorème central limite et la convergence uniforme pour les estimateurs proposés sous certaines conditions. Nous prouvons ces résultats théoriques par une étude de simulation. Enn, nous adoptons les algorithmes d'approximation stochastique pour dénir un estimateur à noyau du mode basé sur l'estimateur récursif de densité à noyau développé par Mokkadem et al. (2009a). En outre, nous établissons sa convergence presque sûre sous l'hypothèse de mélange fort et nous corroborons ces résultats théoriques par des simulations numériques.

Mot-clés: Algorithme d'approximation stochastique, Choix de fenêtre, Consistence forte, Distribution de Pareto, Données mélangeantes, Estimation non paramétrique, Estimateur récursif, Indice de queue, Normalité asymptotique, Valeur extrême. R1, R2, R3, R4) in points x = 0.1, 0.2, 0.3 (1st line), x = 0.4, 0.5, 0.6 (2nd line) and x = 0.7, 0.8, 0.9 (3rd line), and dashed lines represent the values of reference estimator γ(•) in each point x as mentioned above. . . . . . . . 2.3 Qualitative comparaison between the non recursive estimator (2.1.1) and the proposed recursive one (2.2.7). . . . . . . . . . . . . . . . . . . . . . . .

List of Figures

General introduction

Estimation theory has been an attractive area of research that has generated signicant scientic concern and interest among statisticians. It has led to the development of a wide variety of applied elds such as medicine, biology, public health, epidemiology, astronomy, economics and demography. There are three estimation approaches in literature. The rst one stands for the parametric estimation that is the estimation of a nite number of parameter. In this case, the estimators are constructed using either the method of moments, least squared method or maximum likelihood method (See, for instance, [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF] and [START_REF] Mclachlan | Finite mixture models[END_REF]). The second one corresponds to the non parametric estimation that is the estimation of an unknown function from observations. An introduction to non parametric methods in [START_REF] Tsybakov | Recurrent estimation of the mode of a multidimensional distribution[END_REF]. The third estimating model refers to a semi-parametric method which combines both parametric and non parametric aspects. The branch of non parametric estimation has become very attractive in current research. The oldest and most widely used method for non parametric density estimation is the histogram. This method remains insucient to estimate a smooth density. Hence, the introduction of a kernel technique produces a smooth estimation of the probability density function and recties the previous problem. The kernel method of smoothing was introduced by Rosenblatt (1956) and extended by [START_REF] Parzen | On estimation of a probability density function and mode[END_REF]. It was investigated in several directions. For example, the estimation of the density of probability as well as the distribution function, the regression function and the extreme value index function.

The kernel estimators have been improved using stochastic approximation methods. This method has the capacity to facilitate updating estimators when we have new observations. That introduce the notion of the recursivity. As an excellent reference for the stochastic approximation algorithm, we refer the reader to [START_REF] Révész | Robbins-Monro procedure in a Hilbert space and its application in the theory of learning processes I[END_REF][START_REF] Révész | How to apply the method of stochastic approximation in the nonparametric estimation of a regression function[END_REF] and Mokkadem et al. (2009a).

The main objective in this thesis resides in applying the stochastic approximation method to build up a large class of recursive non parametric kernel estimators for dependent and independent variables. This manuscript is structured in terms of four major chapters. In the rst chapter, We provided useful denitions and some asymptotic properties of continuous kernel estimators. Next, we considered various mixing conditions. Subsequently, we introduced the Extreme Value Theory (EVT), we displayed the fundamental theorem in EVT and we recalled certain basic denitions. Additionally, we exhibited the dierent extreme value distributions and we presented dierent estimators of extreme value index as well as recalling their asymptotic properties. Finally, we have described the stochastic Robbins-Monro algorithm which allows us to introduce recursive estimators.

In the second chapter, we applied the stochastic approximation method to dene a class of recursive kernel estimator of the conditional extreme value index. We investigated the properties of the proposed recursive estimator and compared them to those concerning Hill's non recursive kernel estimator. We demonstrated that using some optimal parameters, the proposed recursive estimator dened by the stochastic approximation algorithm proves to be very competitive compared to Hill's non recursive kernel estimator. Finally, the theoretical results are tackled through simulation experiments and illustrated using real dataset about Malaria in Senegalese children. This research work is actually under review after minor revision Ben Khadher and Slaoui (2021c).

Chapter three is an extension of the work of [START_REF] Slaoui | The stochastic approximation method for the estimation of a distribution function[END_REF] to the case of αmixing data. We rst examined the properties of these estimators and compared them to Nadaraya's non recursive distribution estimator. We showed that, using some optimal parameters, the recursive estimators allowed us to obtain quite better results compared to the non recursive distribution estimator under α-mixing condition in terms of estimation error. Then, we elaborated the central limit theorem and the uniform convergence for the proposed estimators under some mild conditions. Finally, we corroborated these theoretical results through a few simulations. This research work was the subject of the following publication Ben Khadher and Slaoui (2021a).

In chapter four, we identied a kernel estimator of the mode based on the recursive kernel density estimator developed by Mokkadem et al. (2009a). In addition, we established its almost sure convergence under strong mixing hypotheses. This research work was the subject of the following accepted paper Ben Khadher and [START_REF] Ben Khadher | Strong consistency of the mode of multivariate recursive kernel density estimator under strong mixing hypothesis[END_REF].

The last part incorporates the closing section which rests upon pertinent concluding remarks as well as promising future perspectives.

Chapter 1

Basic concepts

In this chapter, the readers are provided with a brief review of the scientic background of non parametric estimation. First, generalities on non parametric kernel estimation were displayed. Second, methods allowing to obtain the optimal choice of smoothing parameter estimation were proved. Subsequently, a set of denitions related to types of mixing conditions were integrated. Furthermore, the concept of EVT was introduced. Eventually, the stochastic approximation algorithm which create the recursive estimators was presented.

Non parametric kernel estimation

We are interested in this section in classical non parametric estimation. Notably, the properties of kernel approach of the density, mode and distribution functions are reported. In this section, let X 1 , • • • , X n be independent and identically distributed (iid) R-valued random variables and let f and F denote respectively the probability density and the distribution function of X 1 .

Kernel density estimator

In this subection, we provide some asymptotic properties of continuous kernel density estimators. The rst step is to dene the notion of the kernel.

Denition 1.1.1. A kernel is a function K : R -→ R, which is positive, integrable and satises R K(x)dx = 1.

Some classical examples of kernels function are indicated as follows.

Example 1.1.1.

K (x) = 1 2 1 {|x| 1} (the rectangular kernel), K (x) = (1 -|x|)1 {|x| 1} (the triangular kernel), K (x) = 3
4 (1 -|x| 2 )1 {|x| 1} (the Epanechnikov kernel or the parabolic kernel),

K (x) = 15 16 (1 -|x| 2 ) 2 1 {|x| 1} (the biweight kernel), K (x) = 1 √ 2π exp x 2 2
(the Gaussian kernel).

For further details, we can refer to [START_REF] Tsybakov | Recurrent estimation of the mode of a multidimensional distribution[END_REF]. The well-known kernel density estimator of f was introduced by Rosenblatt (1956)(see also [START_REF] Parzen | On estimation of a probability density function and mode[END_REF]) and dened as

∀x ∈ R, f n (x) = 1 nh n n i=1 K x -X i h n , (1.1.1)
where K : R -→ R is a kernel function and (h n ) is a sequence of positive real numbers that goes to zero as n tends to innity called bandwidth. To investigate the asymptotic behaviors of the estimator (1.1.1), we make the following assumptions:

Assumption 1.1.1.1.

(A1). K(x) = K(-x), ∀ x ∈ R, (A2). R xK(x)dx = 0, (A3). R x 2 K(x)dx < +∞, (A4).
R K 2 (x)dx < +∞. Multiple are the criteria to assess the eciency of the estimator f n . For instance, we established an asymptotic expression for the mean squared error (MSE) of the kernel estimator for any xed value of x. This is dened as

M SE f n (x) = E f n (x) -f (x) 2 .
Developing this expression, we obtain

M SE f n (x) = Bias 2 f n (x) + Var f n (x) ,
where

Bias f n (x) = E f n (x) -f (x),
and

Var f n (x) = E f n (x) -E f n (x) 2 = E f 2 n (x) -E 2 f n (x) .
To evaluate the MSE of f n , we will need to calculate its bias and variance. Assuming that f is bounded, twice dierentiable and f (2) is bounded and substituting that u = x-t hn , we obtain

Bias f n (x) = 1 h n R K (u) [f (x -uh n ) -f (x)] du.
and

Var f n (x) = 1 nh 2 n R K 2 x -t h n f (t)dt - R K x -t h n f (t)dt 2 .
By Taylor series expansion, f (x

-uh n ) = f (x) -uh n f (x) + 1 2 u 2 h 2 n f (2) (x) + o(h 2 n ),
and by applying the properties of the kernel K, we get

Bias f n (x) = 1 2 h 2 n f (2) (x) R z 2 K(z)dz + o(h 2 n )
and

Var f n (x) = 1 nh n f (x) R K 2 (z)dz + o 1 nh n .
The choice of the bandwidth has an important inuence over the quality of kernel estimation. It needs to be carefully determined. The optimal value of h n is obtained by minimizing the asymptotic Mean Integrated Squared Error (M ISE).

M ISE f n = R M SE f n (x) dx = R Bias 2 f n (x) dx + R Var f n (x) dx.
Therefore, this optimal value of h n is expressed by

h opt,n = K 2 2 R t 2 K(t)dt 2 f 2 2 1 5 n -1 5 , (1.1.2)
where . 2 is the Euclidien norm.

Unfortunately, the optimal smoothing parameter (h opt,n ) depends on the unknown quantity

f 2 2 = R f (t) 2 dt.
Hence, it cannot be readily applied in practice. There are several methods to estimate the smoothing parameter. Among the most famous and useful ones are the Plug-in approach and the cross-validation criterion (see [START_REF] Tsybakov | Recurrent estimation of the mode of a multidimensional distribution[END_REF]).

Plug-in:

After dening the theoretical optimal bandwidth as the minimizer of the mean integrated squared error, we estimate the unknown quantities in expression h n . A natural way lies in using non parametric estimator for R f (t) 2 dt. Let us dene

f n (x) = 1 ng 3 n n i=1 K x -X i g n ,
where g n is a prior bandwidth. Several estimators for f 2 2 were developed by [START_REF] Hall | Estimation of integrated squared density derivatives[END_REF]. Thus, they determined the bias corrected estimator in terms of

f 2 2 = f n (x) - 1 ng 5 n K 2 2 .
To obtain an adequate prior bandwidth, [START_REF] Park | Comparison of Data-Driven Bandwidth Selectors[END_REF] set g n as the minimizer for the asymptotic mean squared error of f 2 2 . Using (1.1.2), the prior bandwidth is expressed in function of (h n ) as:

g n = I 1 (K)I 2 (f )h 10/13
n where I 1 (K) contains the fourth derivative as well as convolutions of K, and I 2 (f ) contains the second and third derivatives of f . It follows that, the expression of the optimal bandwith (h n ) is expressed as

h opt,n = K 2 2 R t 2 K(t)dt 2 f 2 2 1/5 n -1/5 ,

Cross-validation:

The usual method for estimating risk is leave-one-out cross-validation. Recall that the risk of

f n is indicated by M ISE f n = E (R) where R f n = R f n (x) -f (x) 2 dx
is the integrated squared error loss function. The loss function, which we now write as a function of smoothing parameter (h n ), (since

f n depend on (h n )) is L(h n ) = R ( f n (x) -f (x)) 2 dx = R f 2 n (x)dx -2 R f n (x)f (x)dx + R f 2 (x)dx.
The last term does not depend on (h n ). As a matter of fact, minimizing the loss is equivalent to minimizing the expected value of

J(h n ) = R f 2 n (x)dx -2 R f n (x)f (x)dx.
We shall refer to E (J(h n )) as the risk, although it diers from the true risk by the constant term R f 2 (x)dx. The cross-validation estimator of risk is represented by

J(h n ) = R f 2 n (x)dx - 2 n n i=1 f (-i) (X i )
where

f (-i) (t) = 1 (n-1)hn n j =i K t-X j hn
is the density estimator obtained after removing the i th observation. Next, the optimization is restricted to a range of values of h n and the one that minimizes J shall be selected.

Kernel mode estimator

The mode is often based on a sequence of the density function f , dened as the value θ which maximizes it, as expressed as follows

f (θ) = sup t∈R f (t).
The kernel estimator of the mode θ is dened as the random variable θ n maximizing the estimator f n (dened in (1.1.1)), which is expressed as

θ n := arg max x∈R f n (x).
The majority of properties of mode estimators are related to those of density estimators and have been explored by several authors. The weak consistency of the kernel sample mode was investigated by [START_REF] Parzen | On estimation of a probability density function and mode[END_REF]. More precisely, it is assumed that the true probability density function f (x) is uniformly continuous in x and that the mode θ is unique. Then he reported the following theorem Theorem 1.1.1. [P arzen (1962)

] If (h n ) is a function of n satisfying lim n→∞ nh 2 n = ∞,
and if the probability density f (x) is uniformly continuous, then for every > 0

P sup -∞<x<∞ |f n (x) -f (x)| < -→ 1, as n -→ +∞.
If {θ n } are the sample modes, and if the population mode is unique, then for every > 0

P [|θ n -θ| < ] -→ 1, as n -→ +∞.
This result was extended in several directions. We can mention for example [START_REF] Cherno | Estimation of the mode[END_REF], [START_REF] Eddy | Optimum kernel estimators of the mode[END_REF][START_REF] Eddy | The asymptotic distributions of kernel estimators of the mode[END_REF] and [START_REF] Vieu | A note on density mode function[END_REF].

The strong consistency was explored by [START_REF] Nadaraya | On nonparametric estimates of density functions and regression curves[END_REF] and [START_REF] Van Ryzin | On strong consistency of density estimates[END_REF]. We recall the following theorem: Theorem 1.1.2. [N adaraya (1965)] We assume that 1. K is a continuous and bounded function such that K(x) -→ 0 as |x| -→ +∞,

The series

∞ n=1 exp -γnh -2 n converges for any γ > 0 where nh -2 n -→ 0 as n -→ +∞, 3. f (x) is uniformly continuous,
then if the mode is unique, the sample mode θ n converges to θ almost surely (a.s.).

The asymptotic normality of kernel estimate of the mode was elaborated by [START_REF] Romano | On weak convergence and optimality of kernel density estimates of the mode[END_REF]. The multidimensional study of the mode was carried out by [START_REF] Samanta | Nonparametric estimation of the mode of a multivariate density[END_REF] and [START_REF] Konakov | On the asymptotic normality of the mode of multidimensional distributions[END_REF].

Kernel distribution estimator

There has been a considerable development of methods for smooth estimation of distribution functions. The most popular one is the kernel approach which is identied by [START_REF] Nadaraya | Some New Estimates for Distribution Functions[END_REF] as follows

F n (x) = 1 n n i=1 K x -X i h n (1.1.3)
where

K (z) = z ∞ K(x)dx.
In the following, we assume that the kernel function K satises (A1)-(A3) in (1.1.1.1). We are now ready to state the basic properties of the kernel distribution estimator (3.1.3). In order to measure the quality of our proposed estimator F n (x), we use the following quantity:

M W ISE F n = E R F n (x) -F (x) 2 f (x)dx = R Bias F n (x) 2 f (x)dx + R Var F n (x) f (x)dx.
Assuming that f is bounded, dierentiable and f is bounded, the bias and the variance of Nadaraya's estimator F n (x) are given by:

Bias F n (x) = 1 2 h 2 n f (x)µ 2 (K) + o(h 2 n ), Var F n (x) = 1 n F (x)(1 -F (x)) - h n n f (x)φ(K) + o h n n ,
where

µ 2 (K) = R t 2 K(t)dt and φ(K) = 2 R tK(t)K(t)dt.
It follows that

M W ISE F n = 1 n V F - h n n I 1 φ(K) + 1 4 h 4 n I 2 µ 2 2 (K) + o(h 4 n ),
where

I 1 = R f 2 (x)dx, I 2 = R (f (x)) 2 f (x)dx and V F = R F (x)(1 -F (x))f (x)dx.
To minimize the M W ISE F n , the bandwidth (h n ) must be equal to

h n = I 1 φ(K) I 2 µ 2 2 (K) 1/3 n -1/3 ,
and then we get

M W ISE F n = n -1 V F 1 - 3 4
I 4/3 1 Θ(K) I 1/3 2 V F n -1/3 + o(n -1/3 ) ,
where

Θ(K) = φ(K) 4 µ 2 2 (K) 1/3
.

The properties of Fn have been investigated by several authors. The uniform convergence was elaborated by [START_REF] Nadaraya | Some New Estimates for Distribution Functions[END_REF], [START_REF] Winter | Strong uniform consistency of integrals of density estimators[END_REF], [START_REF] Yamato | Uniform convergence of an estimator of a distribution function[END_REF] and [START_REF] Singh | Nonparametric estimates of distribution functions[END_REF]. The asymptotic normality was addressed by Watson and Leadletter (1964).

1.2

Mixing conditions

Numerous probabilistic tools have been developed for measuring the dependence between variables. For a nite-variance process, elementary measures of dependence are the autocovariances and autocorrelations. Mixing assumptions, introduced by Rosenblatt (1956), are used to convey dierent ideas of asymptotic independence between the past and future processes. We dene now the popular α-mixing coecient.

Denition 1.2.1. Let X = (X i ) i 1 be a sequence of random variables. Given a positive integer n, set

α (n) = sup k | P (A ∩ B) -P (A) P (B) |, A ∈ F k 1 (X) and B ∈ F ∞ k+n (X) , (1.2.1)
where

F k i (X) is the σ-eld of events generated by X j , i j k. The sequence is α-mixing if the mixing coecient α (n) -→ 0 as n -→ ∞.
The α-mixing, called also the strong mixing, condition was introduced by Rosenblatt (1956). It is the weakest among the known mixing conditions in the literature. There are numerous examples of stochastic processes satisfying the α-mixing condition, such as ARMA processes, the threshold extension, the EXPAR model, the simple ARCH models, their GARCH extension and the bilinear Markovian models. If the mixing condition α(n) = O (exp -an ) for some a > 0, the process is exponentially strongly mixing, where a is the mixing rate and 1/a is the mixing time. The process is geometrically strongly mixing when there exists ρ ∈ (0, 1) such that, α(n

) = O (ρ n ). Then, if α(n) = O n -k
for some k > 0, the process is polynomially strongly mixing. The α-mixing has many practical applications (see [START_REF] Doukhan | Mixing: properties and examples[END_REF], [START_REF] Bosq | Nonparametric statistics for stochastic processes: estimation and prediction[END_REF], [START_REF] Bradley | Introduction to strong mixing conditions[END_REF] and [START_REF] Dedecker | Weak dependence with examples and applications[END_REF] for more details).

There exist various other mixing conditions used in the literature. We mention, for instance, the β-mixing condition (see [START_REF] Kolmogorov | Uber die analytischen methoden inder wahrschein lichkeitsrechnung[END_REF]), φ-mixing condition (see [START_REF] Ibragimov | Some limit theorems from stationary processes[END_REF]), ψ-mixing condition (see [START_REF] Blum | On the strong law of large numbers for a class of stochastic processes[END_REF]) and ρ-mixing condition (see [START_REF] Hirschfeld | A connection between correlation and contingency[END_REF]).

Extreme Value Theory

The asymptotic theory of sample extremes has been developed in parallel with the central limit theory, and in fact both theories bear to a certain extend some resemblance. Let X 1 , X 2 ,. . . , X n be independent and identically distributed random variables having a common distribution function F . The central limit theory concerns the limit behavior of the sum X 1 + X 2 + . . . + X n as n -→ ∞, whereas the theory of sample extremes concerns the asymptotic behavior of the sample extremes max(X 1 , X 2 , . . . , X n ) or min(X 1 , X 2 , . . . , X n ) as n -→ ∞.

In this thesis, we shall consider the maxima of the sample, Knowing that all results obtained can be easy reformulated for sample minima according to the following formula:

min(X 1 , X 2 , . . . , X n ) = -max(-X 1 , -X 2 , . . . , -X n ).

Extreme value distributions

The main result in EVT was introduced by [START_REF] Fisher | Limiting forms of the frequency distribution of the largest or smallest member of a sample[END_REF] and [START_REF] Gnedenko | Sur la distribution limite du terme maximum d'une série aléatoire[END_REF]. They proved that the distribution of the extreme values of an iid sample from a cumulative distribution function F can converge only to one distribution from the three possible ones.

Theorem 1. 3.1. [F isherandT ippet (1928); [START_REF] Gnedenko | Sur la distribution limite du terme maximum d'une série aléatoire[END_REF]] Under certain regularity conditions on the distribution function F , there exist a real parameter γ and two normalizing series

(a n ) n≥1 ⊂ R * + and (b n ) n≥1 ⊂ R such that for all x ∈ R, lim n→∞ P max (X 1 , • • • , X n ) -b n a n ≤ x -→ n→+∞ H γ (x),
with,

if γ > 0, H γ (x) =    0 if x < 0 exp -x -1 γ if x 0 if γ < 0, H γ (x) =    exp -(-x) -1 γ if x < 0 1 if x 0 if γ = 0, H 0 (x) = exp [exp (-x)] for all x ∈ R.
The distribution function H γ is called extreme value distribution. It is indexed by a shape parameter γ called the extreme value index. This parameter accounts for the behaviour of the tail of the distribution.

Three domains of attraction, depending on the sign of γ , should be distinguished:

• If γ > 0, F is said to belong to the Fréchet domain of attraction. This domain includes distribution with heavy tails, i.e. their survival distribution function decreases as a power function.

• If γ = 0, F is said to belong to the Gumbel domain of attraction. This domain includes distributions with light tails, i.e. their survival distribution function decreases as an exponential rate.

• If γ < 0, F is said to belong to the Weibull domain attraction. This domain includes distributions with short tails, i.e. they have a nite endpoint x F = inf {x, F (x) 1}.

Remark 1.

F (x) = 1 -(x/a) -α , for x > a > 0 and α > 0. Let a n = aα -1 n 1/α , b n = an 1/α and extreme value index γ = 1/α > 0, P max (X 1 , • • • , X n ) -b n a n ≤ x = F n (a n x + b n ) = 1 - (1 + x/α) -α n n , So P max(X 1 ,••• ,Xn)-bn an ≤ x -→ exp -(1 + x/α) -α = H 1/α (x) as n -→ ∞ for x > -α.

Characterization of the domains of attraction

The characterization of domains of attraction relies on the theory of regularly-varying functions.

Denition 1.3.1. A positive function L is regularly-Vaying with index δ ∈ R at innity

if lim x→∞ L(tx) L(x) = t δ , ∀ t > 0.
This property is denoted by L ∈ RV δ . If δ = 0, the function L is said to be slowly-varying.

A well known example of a slowly-varying function is L(x) = ln x.

Let us now display the expressions of the distribution function in each domain.

• Fréchet Domain of attraction Theorem 1.3.1. F is in the domain of attraction of a Fréchet distribution with shape parameter ξ if and only if F is regularly varying with index -1/ξ ie:

F (x) = 1 -F (x) = x -1/ξ L(x), x > 0,
where L is a slowly-varying function.

< x F ≤ ∞ such that F (x) = c(x) exp - x x 0 g(t) a(t) dt ,
where a, c and g three functions verifying a (x) -→ 0, c(x) -→ c > 0 and g(x) -→ 1 as x -→ x F .

Estimation of the extreme value index

The extreme value index γ plays a central role in terms of searching for the shape of the distribution tail. We need to estimate it in order to better understand the nature of the studied extreme distribution. Several methods for estimating this parameter are proposed in the literature of EVT. The Hill estimator (see [START_REF] Hill | A simple general approach to inference about the tail of a distribution[END_REF]), the Pickands estimator (see [START_REF] Pickands | Statistical inference using extreme order statistics[END_REF]) and the Dekkers et al. moment estimator (see [START_REF] Dekkers | A moment estimator for the index of an extreme-value distribution[END_REF]) present the most widely used ones in practice. An extensive discussion of estimation methods for EVT models can be found in [START_REF] Embrechts | Modelling Extremal Events[END_REF]. We recall below the three most frequently used estimators of the extreme value index and their asymptotic properties.

The Hill estimator:

The Hill estimator is dened by:

γ H tn (x) = 1 t n tn i=1 i (log X n-i+1,n -log X n-i,n ) = 1 t n tn i=1 log X n-i+1,n -log X n-tn,n ,
where X 1,n ≤ • • • ≤ X n,n are the associated order statistics to the sample X 1 , • • • , X n and t n is the number of the top order statistics (number of extremes) used for the estimation of γ. The construction of this estimator is based on the maximum likelihood method. It is well known that the Hill estimator displays a very good performance, that is competitive with respect to other EVT methods of estimation. Theoretically, the Hill estimator is favorably considered in view of its asymptotic properties, which are summarized in Embrechts et al. (1997)(Theorem 6.4.6):

• Weak consistency: if t n -→ ∞ and t n /n -→ 0 for n -→ ∞, then γ H tn P -→ γ. • Strong consistency: if t n /n -→ ∞ and t n / log log n -→ ∞ for n -→ ∞, then γ H tn a.s.
-→ γ.

• Asymptotic normality: under additional hypotheses,

√ t n γ H tn -γ D -→ N (0, γ 2 ),
where P -→ denotes the convergence in probability,

D

-→ the convergence in distribution and N the gaussian-distribution.

Pickands Estimator:

This estimator is proposed by [START_REF] Pickands | Statistical inference using extreme order statistics[END_REF] to estimate the shape parameter of any of the three extreme value distributions. It is expressed as

γ P tn = 1 log 2 log log X n-tn+1,n -log X n-2tn+1,n X n-2tn+1,n -X n-4tn+1,n .
Their asymptotic properties are well studied in [START_REF] Dekkers | A moment estimator for the index of an extreme-value distribution[END_REF]:

• Weak consistency: if t n -→ ∞ and t n /n -→ 0 for n -→ ∞, then γ P tn P -→ γ. • Strong consistency: if t n /n -→ ∞ and t n / log log n -→ ∞ for n -→ ∞, then γ P tn a.s.
-→ γ.

• Asymptotic normality: under additional hypotheses which can be consulted in [START_REF] Dekkers | A moment estimator for the index of an extreme-value distribution[END_REF],

√ t n γ P tn -γ D -→ N 0, γ 2 (2 2γ+1 + 1) 4 (log 2) 2 (2 γ -1) 2 .

The moment Estimator:

The moment estimator is identied by [START_REF] Dekkers | A moment estimator for the index of an extreme-value distribution[END_REF] and determined as:

γ M tn = M (1) tn + 1 - 1 2    1 - M (1) tn 2 M (2) tn    -1 , 1 < t n < n,
where

M (r) tn = 1 tn tn j=1 log X n j +1,n -log X n-tn,n r , r = 1, 2. Note that M (1)
tn corresponds to the Hill estimator. It is called moment estimator since M

(1) tn can be considered as empirical moments of the order r. Further more, it is known as the Dekkers-Einmahl-de Haan estimator as an extension to the Hill estimator. Note that the asymptotic properties of γ M tn estimator were investigated in [START_REF] Dekkers | A moment estimator for the index of an extreme-value distribution[END_REF].

Suppose F belongs to one of the domain of attraction with γ ∈ R, x F > 0 and let (t n ) n≥1 be a sequence of integers such that 1 ≤ t n < n, t n -→ ∞ and t n /n -→ 0 as n -→ ∞.

• Weak consistency: then, γ M tn P -→ γ.

• Strong consistency: if t n /(log n)δ -→ ∞ as n -→ ∞ for δ > 0, then γ M tn a.s.

-→ γ.

• Asymptotic normality: under additional hypotheses on the distribution function F (see [START_REF] Dekkers | A moment estimator for the index of an extreme-value distribution[END_REF], Theorem 3.1 and Corollary 3.2),

√ t n γ M tn -γ D -→ N 0, σ 2 γ ,
where

σ 2 γ =      1 + γ 2 if γ ≥ 0 (1 -γ) 2 (1 -2γ) (1 -γ + 6γ 2 ) (1 -3γ) (1 -4γ) if γ < 0 . 1.4
Stochastic approximation method and recursive estimators Stochastic algorithms have been widely used in numerous areas including adaptive control, system identication, sequential change detection and transmission systems, see [START_REF] Benveniste | Adaptive Algorithm and Stochastic Approximations[END_REF] for multiple interesting examples. The stochastic algorithm method allows us to construct a class of recursive estimators. The advantage of recursive estimators lies in the fact that their update, from a sample of size n to one of size n + 1, requires considerably less computations.

The Stochastic approximation method

The general form of stochastic algorithm is:

θ n = θ n-1 + γ n φ (θ n-1 , W n ) + γ 2 n µ n (θ n-1 , W n ) , (1.4.1)
where (θ n ) stands for the sequence to be recursively updated, (γ n ) corresponds a positive sequence of real numbers decreasing towards zero, (W n ) represents a sequence of random variables representing the on-line observations, φ(θ, W ) refers to the function which essentially denes how the parameter θ is updated as a function of new observation and

µ n (θ n-1 , W n )
relates to a small perturbation on the algorithm. The behavior of this algorithm was investigated by [START_REF] Benveniste | Adaptive Algorithm and Stochastic Approximations[END_REF], the special case when µ n = 0 was considered by [START_REF] Delyon | General results on the convergence of stochastic algorithms[END_REF]. Algorithm (1.4.1) coincides with the one analyzed by [START_REF] Kushner | General convergence results for stochastic approximations via weak convergence theory[END_REF], [START_REF] Ljung | Strong Convergence of a Stochastic Approximation Algorithm[END_REF] and [START_REF] Ruppert | Almost Sure Approximations to the RobbinsMonro and KieferWolfowitz Processes with Dependent Noise[END_REF]:

θ n = θ n-1 + γ n [φ(θ n-1 ) -W n + η n ] , (1.4.2)
where η n stands for a random variables which converges to 0 almost surely and φ corresponds to a measurable unknown function.

They asserted that (1.4.2) includes the [START_REF] Robbins | A Stochastic Approximation Method[END_REF] and [START_REF] Kiefer | Stochastic Estimation of the Maximum of a Regression Function[END_REF] stochastic approximation processes, which allow the search for zero θ of the function φ. The application of RobbinsMonro's procedure to construct a stochastic approximation algorithm was identied by [START_REF] Révész | Robbins-Monro procedure in a Hilbert space and its application in the theory of learning processes I[END_REF][START_REF] Révész | How to apply the method of stochastic approximation in the nonparametric estimation of a regression function[END_REF] and extended by [START_REF] Tsybakov | Recurrent estimation of the mode of a multidimensional distribution[END_REF]. Most of the classical results for the RobbinsMonro and KieferWolfowitz processes require the assumption E [W n |F n-1 ] = 0, where F n-1 stands for the σ-algebra of the events occurring up the time n -1. Under standard conditions on the function φ and on the sequence (γ n ), [START_REF] Kushner | Stochastic approximation and recursive algorithms and applications[END_REF] highlighted that

θ n -→ θ a.s. as n -→ ∞.
In the following subsections, two examples of recursive estimators are established using the Robbins and Monro algorithm (See [START_REF] Robbins | A Stochastic Approximation Method[END_REF]).

Recursive kernel estimators

Recursive kernel density estimator

In order to construct a stochastic algorithm, which approximates the unknown density function f at a given point x, Mokkadem et al. (2009a) dened an algorithm to search for the zero of the function g : y -→ f (x) -y as follows:

(i) f 0 (x) ∈ R, (ii) ∀n 1, we set f n (x) = f n-1 (x) + γ n Z n (x)
, where the stepsize (γ n ) is a sequence of positive real numbers that go to zero and

(Z n ) is a sequence of functions Z n : R → R dened by Z n (x) = g(f n-1 (x)) -W n + η n .
Departing from the fact that E(W n |F n-1 ) = 0, where F n-1 stands for the σ-algebra of the events occurring at the time

n -1, it follows that E(Z n (x)) = f (x) -f n-1 (x) + η n .
Adapting the approach of [START_REF] Révész | Robbins-Monro procedure in a Hilbert space and its application in the theory of learning processes I[END_REF][START_REF] Révész | How to apply the method of stochastic approximation in the nonparametric estimation of a regression function[END_REF] and noting that

E h -d n K(h -1 n (x -X n )) = f (x) + ξ n (x),
where ξ n (x) goes to zero as n goes to innity, we set

Z n (x) = h -d n K(h -1 n (x -X n )) - f n-1 (x).
Therefore, the recursive estimator f n of the density function f at the point x can be stated as

f n (x) = (1 -γ n )f n-1 (x) + γ n K(h -1 n (x -X n )). (1.4.3)
Further more, we suppose that f 0

(x) = 0. Let Π n = n j=1
(1 -γ j ). As a matter of fact, we infer from Equation (1.4.3) that f n can be rewritten as

f n (x) = Π n n k=1 Π -1 k γ k K x -X k h k .

Recursive kernel distribution estimator

In order to construct a stochastic algorithm, which approximates the function F at a given point x, Slaoui (2014b) dened an algorithm to search for the zero of the function h : y → F (x) -y as follows:

(i) we set F 0 (x) ∈ [0, 1].
(ii) For all n ≥ 1, we set

F n (x) = F n-1 (x) + γ n Q n (x),
where the stepsize (γ n ) represents a positive sequence of real numbers decreasing to zero and

(Q n ) determins a sequence of functions Q n : R → R dened by Q n (x) = φ(F n-1 (x)) -W n + η n .
Relying upon the fact that E(W n |F n-1 ) = 0, where F n-1 stands for the σ-algebra of the events occurring up the time n -1, it follows that

E(Q n (x)) = F (x) -F n-1 (x) + η n .
Based on the approach of [START_REF] Révész | Robbins-Monro procedure in a Hilbert space and its application in the theory of learning processes I[END_REF][START_REF] Révész | How to apply the method of stochastic approximation in the nonparametric estimation of a regression function[END_REF] and noting that

E K(h -1 n (x -X n )) = F (x) + ξ n (x)
, where ξ n (x) goes to zero as n goes to innity and

K(z) = z -∞ K(u)du, we set Q n (x) = K(h -1 n (x -X n )) -F n-1 (x).
Hence, the recursive estimator F n of the distribution function F at the point x can be expressed as

F n (x) = (1 -γ n )F n-1 (x) + γ n K(h -1 n (x -X n )). (1.4.4)
Further more, we suppose that

F 0 (x) = 0. Let Π n = n j=1
(1 -γ j ). Thus, we infer from Equation (1.4.4) that F n can be rewritten as

F n (x) = Π n n k=1 Π -1 k γ k K x -X k h k .
Recall that all denitions and theoretical concepts, introduced in this chapter, allows us to better understand the following chapters.

Introduction

The EVT is a branch of statistics which studies the asymptotic distributions of extreme values. It can be a maximum or a minimum of a set of random variables. This theory was developed by Emil Julius [START_REF] Gumbel | Statistics of Extremes[END_REF]. It is widely applied in many research areas like climate changes, environmental risks, insurance and nancial banking (see [START_REF] Beirlant | Statistics of extremes-Theory and applications[END_REF] for a list of interesting examples). Estimation of the tail index, associated with a random variable Y , is one of the main problems in the area of EVT. Therefore, a lot of research, aiming to estimate this parameter, carried out during last decades (see for example [START_REF] Embrechts | Modelling Extremal Events[END_REF], Beirlant et al. (2004), De Haan and[START_REF] De Haan | Extreme Value Theory-An Introduction[END_REF], [START_REF] Reiss | Statistical Analysis of Extreme Values[END_REF], [START_REF] Gardes | A moving window approach for nonparametric estimation of the conditional tail index[END_REF], [START_REF] Gardes | Functional nonparametric estimation of conditional extreme quantiles[END_REF] and [START_REF] Stuper | A moment estimator for the conditional extreme-value index[END_REF]). We denote by γ the tail index which characterizes the distribution tail heaviness of Y . For example when γ is positive, the survival function of Y decreases polynomially to zero.

It can be estimated parametrically using the [START_REF] Hill | A simple general approach to inference about the tail of a distribution[END_REF] estimator and non parametrically using a kernel version of the Hill's estimator proposed by [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF]. Improved approaches have recently appeared in literature. Among them, we can mention [START_REF] Beirlant | Statistics of extremes-Theory and applications[END_REF] to proposed the generalized Hill estimator; [START_REF] Brilhante | A simple generalisation of the Hill estimator[END_REF] to dened a moment of order p estimator which reduces to the Hill estimator for p = 0; Beran et al. ( 2014) who proposed a harmonic moment tail index estimator; Paulauskas andVaieciulis (2013, 2017) who elaborated parametric families of functions of the order statistics.

In many practical applications, it is often the case that the variable of interest Y can be linked to a covariate X. In this case, the extreme-value index of the conditional distribution of Y given X = x can depend on x; the problem is then to estimate the conditional extreme-value index x -→ γ(x). Motivating examples in the literature include the estimation of the maximal production level as a function of the quantity of labor (see [START_REF] Daouia | Frontier estimation and extreme value theory[END_REF]), studying extreme temperatures as a function of various topological parameters (see [START_REF] Ferrez | Extreme temperature analysis under forest 400 cover compared to an open eld[END_REF], or analyzing extreme earthquakes as a function of the location (see [START_REF] Pisarenko | Characterization of the frequency of extreme earthquake events by the generalized Pareto distribution[END_REF]

). Let (X i , Y i ), i = 1, • • • , n, be independent realizations of the random vectors (X, Y ) ∈ R d ×R + 0
, where X is a d-dimensional covariate with joint density function g, d 1. The probability density function of Y given X = x is dened as f (y|x) = P(Y = y|X = x) and the conditional survival function of Y given X = x is denoted by F (y|x) = P(Y > y|X = x). Now, we dene the kernel version of Hill's estimator of the conditional extreme value index proposed by [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF], which is expressed as follows

γn (x) = 1 n n i=1 K hn (x -X i ) [ln Y i -ln t n ] 1 {Y i >tn} 1 n n i=1 K hn (x -X i ) 1 {Y i >tn} , (2.1.1)
where

K h (x) := h -d K (h -1 x) with K is a kernel function and (t n ) is a nonrandom thresh- old sequence tending to ∞ as n -→ ∞.
Recently, recursive estimation has drawn the attention and whetted the interest of multiple researchers. Recursivity means that the estimator calculated from the rst n observations, say θ n , is a function of θ n-1 . More precisely, we can easily update the estimator value with each additional observation specialy in large sample sizes. The basic objective of the present work lies in applying the stochastic approximation method to construct a recursive kernel estimator of the conditional extreme value index dened in (2.1.1). To the best of our knowledge, this tail index estimator construction was not previously considered in literature and it aims to improve the estimation accuracy. It turns out that this estimator depends on two important parameters, which are the bandwidth and the stepsize of the stochastic algorithm. By making an adequate choice of the two parameters, the proposed recursive estimator can be very competitive to Hill's non recursive kernel estimator in terms of estimation error and much better in terms of computational costs.

The remainder of the chapter is organized as follows. In Section 2, we identify our estimator and we set forward its asymptotic properties. Simulation experiments and investigation of real data are presented in Section 3. Finally, the last section wraps up the conclusion and provides new perspectives for future works.

2.2

Construction of the estimator and asymptotic prop-

F (y|x) -υ -1 γ(x) b(y|x) = υ -1 γ(x) υ ρ(x) γ(x) -1 ρ(x)γ(x)
.

Additional conditions are needed for ensuring the asymptotic properties of the estimators. Let d(x, y) denote the Euclidean distance between x and y, for all x, y ∈ R d .

(C4): There exists c g > 0 such that for all x, y ∈ R d ,

|g (x) -g(y)| ≤ c g d(x, y).
(C5): There exists c F > 0 and y 0 > 1 such that for all x, z ∈ R d ,

sup y≥y 0 ln F (y|x) ln F (y|z) -1 ≤ c F d(x, z).
Moreover, we impose a condition on the kernel function K.

(C6): K is a bounded density function on R d , with support Ω included in the unit hypersphere of R d .

Our idea rests upon to construct a recursive estimator for the conditional tail index γ(x). This recursive version is based on the estimator proposed by [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF] which is a rational function. Therefore, it will be presented as a ratio of two estimators a n (x) and b n (x). The denominator b n (x) is an estimator of the function b(x) = g (x) F (t n |x). The nominator a n (x) is an estimator of the function

a(x) = γ(x)F (t n |x)C x g (x)
, where

C x = 1 + b(t n |x) γ(x)ρ(x) 1 1 -ρ(x) -1 + r n,x
and (r n,x ) is a non-random sequence, tending to 0 as n -→ ∞, dened as

r n,x = ρ(x) γ 2 (x) ∞ 1 z -1 γ(x) -1   γ 2 (x) z 1 γ(x) F (tnz|x) F (tn|x) -1 b(t n |x) - z ρ(x) γ(x) -1 ρ(x)γ(x)   dz.
Remark 2.2.1. Since C x tends to 1, we can remove it safely from the expression of a(x).

Thus it can be written as a(x) = γ(x)F (t n |x)g (x) .

Remark 2.2.2. Based on a deterministic threshold as in the article [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF] and [START_REF] Ndao | Nonparametric estimation of the conditional extreme value index with random covariates and censoring[END_REF], we use the deterministic threshold. It is possible also to consider a random threshold (t n ) as in the article of [START_REF] Stuper | A moment estimator for the conditional extreme-value index[END_REF]. Additionally, we can even make the comparison between two results.

Construction of a recursive estimator of the function a (x):

Let us introduce the stochastic algorithm to estimate the function a(•) at a point x. It is based on searching the zero of the function f 1 : y -→ a(x)-y. Following Robbins-Monro's procedure, this algorithm is dened as follows:

(i) a 0 (x) ∈ R, (ii) ∀n 1, we set a n (x) = a n-1 (x) + γ n Z n (x)
, where the stepsize (γ n ) is a sequence of positive real numbers that goes to zero and Z n (x) is a sequence of function

Z n : R d → R dened by Z n (x) = f 1 (a n-1 (x)) -W n + ζ n ,
where ζ n is a random variables converges to 0 almost surely.

To construct W n (x), we follow the approach of [START_REF] Révész | Robbins-Monro procedure in a Hilbert space and its application in the theory of learning processes I[END_REF][START_REF] Révész | How to apply the method of stochastic approximation in the nonparametric estimation of a regression function[END_REF], [START_REF] Tsybakov | Recurrent estimation of the mode of a multidimensional distribution[END_REF] and [START_REF] Slaoui | Large and moderate principles for recursive kernel density estimators dened by stochastic approximation method[END_REF]Slaoui ( , 2014aSlaoui ( ,b, 2018) ) which are based on the classical property of stochastic algorithms (which is E [W n (x)|F n-1 ] = 0, where F n-1 stands for the σ -algebra of the events occurring at the time n -1). Then, it comes E (Z n (x)) = a(x) -a n-1 (x) + ζ n . In addition, we introduce a kernel K (which is a function satisfying R d K(z)dz = 1), and a bandwidth (h n ) (which is a sequence of positive real numbers that goes to zero when n -→ ∞) . Moreover, we have

E K hn (x -X n ) [ln Y n -ln t n ] 1 {Yn>tn} = a(x) + η n (x)
, where η n (x) goes to zero as n goes to ∞. Then, we set

Z n (x) = K hn (x -X n ) [ln Y n -ln t n ] 1 {Yn>tn} -a n-1 (x).
The stochastic approximation algorithm introduced in Mokkadem et al. (2009a) which estimates recursively the function a at the point x is dened as follows:

a n (x) = (1 -γ n )a n-1 (x) + γ n K hn (x -X n ) [ln Y n -ln t n ] 1 {Yn>tn} . (2.2.1)
Considering a 0 (x) = 0, the estimator a n dened in (2.2.1) can be rewritten as

a n (x) = Π n n k=1 Π -1 k γ k K h k (x -X k ) [ln Y k -ln t n ] 1 {Y k >tn} , (2.2.2)
with

Π n = n k=1 (1 -γ k ). (2.2.3)
Construction of a recursive estimator of the function b (x):

We apply the stochastic algorithm to estimate the function b(•) at a point x. It is based on searching the zero of the function f 2 : y -→ b(x) -y. Following Robbins-Monro's procedure, this algorithm is dened as follows:

(i) b 0 (x) ∈ R, (ii) ∀n 1, we set b n (x) = b n-1 (x) + β n T n (x)
, where the stepsize (β n ) is a sequence of positive real numbers that goes to zero and T n (x) is an observation of the function

f 2 at the point b n-1 (x).
Based on the same previously used approach, we consider

T n (x) = K hn (x -X n ) 1 {Yn>tn} - b n-1 (x)
, with the same bandwidth (h n ) and kernel function K h previously dened. Then, the stochastic approximation algorithm to estimate recursively the function b at the point

x is dened as follows:

b n (x) = (1 -β n )b n-1 (x) + β n K hn (x -X n ) 1 {Yn>tn} . (2.2.4)
Considering b 0 (x) = 0, the estimator b n dened by (2.2.4) can be rewritten as

b n (x) = Q n n k=1 Q -1 k β k K h k (x -X k ) 1 {Y k >tn} , (2.2.5) with Q n = n k=1 (1 -β k ). (2.2.6)
Then, our proposed recursive estimator for the conditional tail index γ(x) is dened as:

γ n (x) := a n (x) b n (x) = Π n n k=1 Π -1 k γ k K h k (x -X k ) [ln Y k -ln t n ] 1 {Y k >tn} Q n n k=1 Q -1 k β k K h k (x -X k ) 1 {Y k >tn} . (2.2.7)
The second objective of our chapter is to study the properties of the recursive estimator dened by (2.2.7) and to compare them with the kernel version of Hill's estimator of the conditional extreme value index dened in (2.1.1).

The asymptotic properties of γ n are investigated in the next subsection.

Asymptotic results

In order to obtain the bias and the variance of the recursive estimator γ n dened by (2.2.7), we rst calculate those of the recursive estimator a n dened by (2.2.2). Then, we calculate those of the recursive estimator b n dened by (2.2.5). Throughout this chapter, stepsizes and bandwidths are considered to belong to the following regularly varying sequences class.

Denition 2.2.1. Let u ∈ R and (u n ) n 1 be a nonrandom positive sequence. We say that

u n ∈ GS(u) if lim n→∞ n 1 - u n-1 u n = u.
This condition was introduced by [START_REF] Galambos | Regularly Varying Sequences[END_REF]. The acronym GS stands for (Galambos and Seneta). Typical sequences in GS(u) are, for b ∈ R, n u (log n) b , n u (log log n) b and so on. Finally, we impose the following additional conditions:

(C7):

(i) γ n ∈ GS(-α) with α ∈ (1/2, 1]. (ii) h n ∈ GS(-p) with p ∈ (0, α/d). (iii) lim n→∞ nγ n ∈ (min (p, (α -pd) /2) , ∞]. (iv) β n ∈ GS(-b) with b ∈ (1/2, 1]. (v) lim n→∞ nβ n ∈ (min (p, (b -pd) /2) , ∞]. (vi) nh d+2 n ln 2 t n -→ n→∞ ∞.
The following notations will be often used in this chapter:

ε = lim n→∞ (nγ n ) -1 . (2.2.8) ε 1 = lim n→∞ (nβ n ) -1 . (2.2.9) c F = c F z 2 such as z ∈ B * R d (0, 1) = {x ∈ R d ; 0 < x 2 ≤ 1}. C = - 1 γ(x) + o(1) c F u 2 for all u ∈ Ω. m n (x) = E K hn (x -X n )(ln Y n -ln t n )1 {Yn>tn} . m n (x) = E (ln Y n -ln t n )1 {Yn>tn} |X n = x .
Since we are interested in the asymptotic behavior of the estimator γ n , we shall start by giving the asymptotic behavior of the estimator a n .

Theorem 2.2.1. (Bias and variance of the estimator a n )

Let Assumptions (C1) -(C7) hold.

1. If p ∈ (0, α/ (d + 2)], then E(a n (x)) = a(x) + O(h n ln t n ). (2.2.10) If p ∈ (α/ (d + 2) , 1/d), then E(a n (x)) = a(x) + O( γ n h -d n ). (2.2.11) 2. If p ∈ (0, α/ (d + 2)), then Var(a n (x)) = o(h 2 n ln 2 t n ).
(2.2.12)

If p ∈ [α/ (d + 2) , 1/d), then Var(a n (x)) = 6 2 -(α -pd)ε K 2 1 g (x) γ 2 (x)F (t n |x)γ n h -d n +o(γ n h -d n ).
(2.2.13) Departing from the above, we infer that the bias and the variance of the estimator a n heavily depend on the choice of the stepsize (γ n ). We consider an example of choices of (γ n )based on the minimization of the variance.

Choices of (γ n ) minimizing the variance of the estimator a n :

As mentioned in Mokkadem et al. (2009a), by considering the point of view of estimation by condence intervals, it is recommended to minimize the variance of the proposed estimator for condence interval estimation (see also [START_REF] Hall | Eect of bias estimation on coverage accuracy of bootstrap condence intervals for a probability density[END_REF]).

Corollary 2.2.1. Let the assumptions of Theorem 2.2.1 hold. To minimize the asymptotic variance of the estimator a n , α must be chosen equal to 1, (γ n ) n must satisfy lim n→∞ nγ n = 1 -pd, and we then have

Var(a n (x)) = 6 (1 -pd) K 2 1 g (x) γ 2 (x) F (t n |x) nh d n + o 1 nh d n .
The proof of Corollary 2.2.1 follows immediately from (2.2.13).

The following proposition provides the M ISE of the estimator a n . First, we have

M ISE(a n ) = R d M SE(a n (x))dx = R d (E(a n (x)) -a(x)) 2 + Var(a n (x)) dx. Proposition 2.2.1. Let Assumptions (C1)-(C7) hold. 1. If p ∈ (0, α/ (d + 2)), M ISE(a n ) = O(h 2 n ln 2 t n ). 2. If p = α/ (d + 2), M ISE(a n ) = 6 2 -(α -pd)ε K 2 1 R d g (x) γ 2 (x)F (t n |x)dxγ n h -d n +o γ n h -d n + O h 2 n ln 2 t n . 3. If p ∈ (α/ (d + 2) , 1/d), M ISE(a n ) = 6 2 -(α -pd)ε K 2 1 R d g (x) γ 2 (x)F (t n |x)dxγ n h -d n + O(γ n h -d n ).
Now, we treat the asymptotic behavior of the estimator b n , in order to deduce the one of the estimator γ n .

Theorem 2.2.2. (Bias and variance of the estimator

b n ) Let Assumptions (C1)-(C7) hold. 1. If p ∈ (0, b/ (d + 2)], then E(b n (x)) = b(x) + O(h n ln t n ).
(2.2.14)

If p ∈ (b/ (d + 2) , 1/d), then E(b n (x)) = b(x) + O β n h d n . (2.2.15) 2. If p ∈ (0, b/ (d + 2)), then Var(b n (x)) = O(h 2 n ln 2 t n ). (2.2.16) If p ∈ [b/ (d + 2) , 1/d), then Var(b n (x)) = 1 2 -(b -pd)ε 1 K 2 2 g (x) F (t n |x) β n h d n + O β n h d n .
(2.2.17)

The bias and the variance of the estimator b n dened by the stochastic approximation algorithm (2.2.4), then heavily depend on the choice of the stepsize (β n ). For an adequate choice, we consider an example of choices of (β n ) based on the minimization of the variance.

Choices of (β n ) n minimizing the variance of the estimator b n :

As mentioned in Mokkadem et al. (2009a), it is recommended to minimize the variance of the proposed estimator for condence interval estimation.

Var(b n (x)) = (1 -pd) K 2 2 g (x) F (t n |x) nh d n + O 1 nh d n .
The proof of Corollary 2.2.2 follows immediately from (2.2.17).

The following proposition provides the M ISE of the estimator b n .

Proposition 2.2.2. Let Assumptions (C1)-(C7) hold.

1. If p ∈ (0, b/ (d + 2)), M ISE(b n ) = O(h 2 n ln 2 t n ).

If

p = b/ (d + 2), M ISE(b n ) = 1 2 -(b -pd)ε 1 K 2 2 R d g (x) F (t n |x)dx β n h d n + O h 2 n ln 2 t n + O β n h d n . 3. If p ∈ (b/ (d + 2) , 1/d), M ISE(b n ) = 1 2 -(b -pd)ε 1 K 2 2 R d g (x) F (t n |x)dx β n h d n + O β n h d n .
Now we present the bias and the variance of γ n .

Theorem 2.2.3. (Bias and variance of γ n )

Let Assumptions (C1)-(C7) hold, and suppose that the stepsize

(β n ) n = (n -1 ) n . 1. If p ∈ (0, α/ (d + 2)], then E ( γ n (x)) -γ (x) = O (h n ln t n ) . (2.2.18) If p ∈ (α/ (d + 2)), 1/d), then E ( γ n (x)) -γ (x) = O γ n h d n . (2.2.19) 2. If p ∈ (0, α/ (d + 2)), then Var ( γ n (x)) = o h 2 n ln 2 t n . (2.2.20) If p ∈ [α/ (d + 2) , 1/d), then Var ( γ n (x)) = 1 b 2 (x) 6 2 -(α -pd)ε K 2 1 g (x) γ 2 (x)F (t n |x) γ n h d n + o γ n h d n . (2.2.21)
Clearly, the bias and the variance of the estimator γ n depend on the choice of the two stepsizes (γ n ) n and (β n ) n .

Let us state now the following Theorem, which gives the weak convergence rate of the proposed recursive estimator γ n dened in (2.2.7) in the special case of (β n ) n = (n -1 ) n . Theorem 2.2.4. Let Assumptions (C1)-(C7) hold, and suppose that (β n ) n = (n -1 ) n .

1. If there exists r > 0 such that

F (t n |x) -1 γ -1 n h d+2 n ln 2 t n -→ n→∞ r then F (t n |x) -1 γ -1 n h d n ( γ n (x) -γ(x)) D -→ N √ rB(x), Var(x) ,
where

B(x) = - C 1 -pε + C 1 -p γ (x) , Var(x) = 1 b 2 (x) 6 2 -(α -pd)ε K 2 1 g (x) γ 2 (x). 2. If F (t n |x) -1 γ -1 n h d+2 n ln 2 t n -→ n→∞ ∞, then 1 h n ln t n ( γ n (x) -γ(x)) P -→ B(x),
Corollary 2.2.3. Under the same assumptions as the previous theorem and if r = 0 then

F (t n |x) -1 γ -1 n h d n ( γ n (x) -γ(x)) D -→ N (0, Var(x)) ,
with

Var(x) = 1 b 2 (x) 6 2 -(α -pd)ε K 2 1 g (x) γ 2 (x).
We can consider the case when the stepsize (β n ) is chosen to minimise the variance of the estimator b n . Similarly, we can obtain the weak convergence rate of the estimator γ n .

The following corollary is a consequence of the previous theorem which gives an asymptotic condence interval of the index function γ.

Corollary 2.2.4. The asymptotic 100(1 -α)% condence interval for γ(x) is given by

γ n + F (t n |x)γ n h -d n √ rB(x) ± u 1-α 2 F (t n |x)γ n h -d n Var(x) ,
where u 1-α 2 is the normal 1 -α 2 quantile.

Simulation study

The target of our applications is to compare the performance of the proposed recursive kernel estimator of the conditional extreme value index given in (2.2.7) to that of Hill's non recursive estimator dened in (2.1.1) using the "Leave One Out" cross-validation bandwidth selection.

2.3.1

The study design

We use the following simulation design: we consider the unidimensional case d = 1 and we simulate N = 500 samples of size n (n = 50, 250) of independent replicates (X i , Y i ) where

X is uniformly distributed on [0, 1] and the conditional distribution of Y i given X i = x is Pareto with parameter γ(x) = 0.5(0.1 + sin(πx) × (1.1 -0.5 exp(-64(x -0.5) 2 ))) (this function was proposed by [START_REF] Daouia | Kernel estimators of extreme level curves[END_REF], it was also used in [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF] and in [START_REF] Ndao | Nonparametric estimation of the conditional extreme value index with random covariates and censoring[END_REF]. The pattern of γ is given in Figure 2.1.

For each of the N simulated samples, we estimate γ(•) at x = (0.1, 0.2, 0.3, • • • , 0.8, 0.9) using the estimator (2.2.7) with a biquadratic kernel K

(x) = 15 16 (1 -x 2 ) 2 1 [-1,1]
. As mentioned in previous papers (see Slaoui (2014a,b)), there is no big inuence on the choice of the kernel K in our setup when the observations are not contamined. In order to calculate our estimator, we need to choose the bandwidth (h n ) n and the threshold (t n ) n . We take t n to be the (n -k) th order statistic Y (n-k) as is usual in extreme value statistics. Moreover, we propose an algorithm for choosing (h n , k). This algorithm adapted from [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF], was considered recently by [START_REF] Ndao | Nonparametric estimation of the conditional extreme value index with random covariates and censoring[END_REF]. The purpose is then to select the bandwidth (h n ) using the following cross-validation criterion

h cv = arg min h∈H n i=1 n j=1 1 {Y i Y j } -F n,-i (Y j |X i ) 2 ,
where

H = {h n = cn -v ; n 1 and (c, v) ∈ {0.1, 0.2, • • • , 0.9}} is a grid of values for (h n ) and F n,-i (y|x) := n j=1,j =i K h (x -X j )1 {Y j y} n j=1,j =i K h (x -X j )
. This criterion was introduced in Yao (1999), implemented by [START_REF] Gannoun | Reference ranges based on nonparametric quantile regression[END_REF], and established in an extreme value context by [START_REF] Daouia | Kernel estimators of extreme level curves[END_REF][START_REF] Daouia | On kernel smoothing for extremal quantile regression[END_REF], [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF] and [START_REF] Ndao | Nonparametric estimation of the conditional extreme value index with random covariates and censoring[END_REF]. Using this bandwidth selection, we consider the following procedure to determine the number of threshold excesses k. This procedure rests on considering for each point x, the following steps:

Step 1: we compute the estimates for

γ n,Y (n-k) (x) with k = 1, • • • , n -1.
Step 2: we construct several successive "blocks" of the estimates γ n,Y (n-k) (x) (one block for k ∈ {1, • • • , 15}, a second block for k ∈ {16, • • • , 30} and so on).

Step 3: we calculate the standard deviation of the estimate in each block.

Step 4: we determine the k-value (denoted by k 1 ) from the block with minimal standard deviation (in particular, we take the median of the k-values in that block).

Finally, we estimate γ(x) by using the estimator γ n (2.2.7) by taking (h n , k) = (h cv , k 1 )

Results

For each conguration of the simulation design parameters (sample size n, stepsize parameters (γ n , β n ) and covariate value x), we calculate the average IAE (Integrated Absolute Error), the average ISE (Integrated Squared Error); and L ∞ of the estimators over N = 500 trials;

IAE = 1 N N i=1 R γ [i] n (x) -γ(x) dx, ISE = 1 N N i=1 R γ [i] n (x) -γ(x) 2 dx and L ∞ = max i=1,••• ,N R γ [i] n (x) -γ(x) dx, where γ [i]
n corresponds to the estimator computed from the ith sample. In order to investigate the comparison estimators, we consider the stepsizes (γ n , β n ) equal to (n -1 , n -1 ), ((2/3) n -1 , n -1 ), (n -1 , (2/3) n -1 ) and ((2/3) n -1 , (2/3) n -1 ) respectively. These four choices of parameters of the recursive estimator are referred to as R1, R2, R3 and R4 respectively. Results are highlighted in Table 2.1. We point out that the major merit of our proposed estimator lies in its update aspect. Indeed, when new sample points are available, it requires less computational cost than non recursive estimator. Moreover, Table 2.1 reveals that our proposed recursive estimator can provide better results in some specic situations that are very close in general to the reference values, which proves the eectiveness of our proposed recursive estimator in terms of the estimation error. Figure 2.2 discloses that all the considered estimators yield good results since the values of γ at each point x ∈ {0.1, 0.2, . . . 0.9} are very close to the median.

Real data application

We considered a Malaria dataset of 176 families in Senegal, totaling 505 children between 2 and 19 years old, living in two villages of Niakhar (Toucar and Diohine). The number of observations was 6986. We measured Plasmodium falciparum Parasite Load (PL) from thick blood smears obtained by nger-prick during two dierent seasons and regularly over a three-year observation period (2001)(2002)(2003). The number of measurements per child ranged from 1 to 15. We refer readers to consult [START_REF] Milet | Genome wide linkage study, using a 250K SNP map, of Plasmodium falciparum infection and mild malaria attack in a Senegalese population[END_REF] for more details about data. These data were used also in Slaoui and Nuel (2014c)

) takes (n -1 , n -1 ), ((2/3) n -1 , n -1 ), (n -1 , (2/3) n -1 ) and ((2/3) n -1 , (2/3) n -1
) respectively. the reference index function, considered by [START_REF] Lekina | Estimation Non-paramétrique des Quantiles Extrêmes Conditionnels[END_REF] and dened as follows:

γ(x i ) = 0.3 x 2 i -min j x 2 j max j x 2 j -min j x 2 j + 0.2,
where x i is the ith value in the data vector. Once a variable change has been taken into consideration, the data vector should be in the same interval [0.2,0.5] proposed by [START_REF] Lekina | Estimation Non-paramétrique des Quantiles Extrêmes Conditionnels[END_REF], so that the previous function γ(x) can be used. Therefore, for any considered estimator γ n of the index function γ, we propose to compute IAE and ISE dened as:

IAE(γ n ) = R |γ n (x) -γ(x)| dx
and

ISE(γ n ) = R (γ n (x) -γ(x)) 2 dx

Conclusion

In this chapter, we tackled the estimation of the conditional extreme value index γ(x) of a heavy-tailed distribution when some random covariate information is available. We elaborate recursive kernel estimator of the extreme value index function based on the stochastic approximation algorithm. The proposed estimator asymptotically follows normal distribution. We subsequently compared the proposed estimator to Hill's non recursive extreme value index estimator. We demonstrated that using some particular stepsizes and a specic bandwidth selection through a cross-validation procedure, the proposed recursive estimator could be very competitive to the non recursive version. Moreover, we highlighted that the proposed estimator is much better in terms of computational costs. Numerical results illustrate the eectiveness of our recursive approach. To this extent, we would state that although our work is an extension of a wealthy historical background, it may be taken further, extended and built upon since it oers dierent perspectives and opens new horizons for future research. We can extend our recursive extreme value index estimator to the case of censored data. We can also propose a new estimator of the conditional extreme quantile using our recursive estimator dened by (2.2.7) and compare it to the classical Weissman estimator. Another direction is to investigate the almost sure convergence and the large and moderate deviation principles of the proposed estimator, which requires non trivial mathematics. This would go well beyond the scope of the present chapter.

Proofs

We introduce the following Lemmas that will enable us to obtain the asymptotic expansion of a n .

Lemma 2.5.1. Let assumption (C3) holds. Then, for t n -→ ∞ as n -→ ∞ we have

m n (x) = γ(x)F (t n |x).
The proof of Lemma 2.5.1 is presented in [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF].

Lemma 2.5.2. Let assumptions (C1)-(C6) hold. Then, for all x ∈ R d such that g (x) > 0 we have for

t n -→ n→∞ ∞ and h n -→ n→∞ 0 with h n ln t n -→ →∞ 0 m n (x) = m n (x)g (x) (1 + O(h n ln t n )) . (2.5.1)
The proof of Lemma 2.5.2 is presented in [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF].

Lemma 2.5.3. Let assumptions (C1) and ( C4)-(C6) hold. Then, for all x ∈ R d such that g (x) > 0, we have for

t n -→ n→∞ ∞ and h n -→ n→∞ 0 with h n ln t n -→ n→∞ 0 E K hn (x -X n ) 1 {Yn>tn} = g (x) F (t n |x) (1 + O(h n ln t n )) . 2.5.1 Proof of Lemma 2.5.3
Since (X i , Y i ), i = 1, • • • , n are independent and identically distributed, we have under the assumption (C6)

E K hn (x -X n ) 1 {Yn>tn} = R d R 1 h d n K x -t h n 1 {y>tn} f (y|t)g(t)dtdy = R d 1 h d n K x -t h n F (t n |t)g(t)dt = Ω K(u)F (t n |x -uh n )g(x -uh n )du.

Now, we consider

E K hn (x -X n ) 1 {Yn>tn} -F (t n |x)g (x) ≤ F (t n |x) Ω K(u) |g(x -h n u) -g (x)| du + F (t n |x) Ω K(u) F (t n |x -uh n ) F (t n |x) -1 g(x -h n u)du := J1 + J2 .
Under the assumption (C5), and since g(x) > 0, we have

J1 ≤ F (t n |x)c g h n Ω u 2 K(u)du = F (t n |x)g (x) O(h n ). (2.5.2)
Concerning J2 , under (C5) and using this equation

F (y|x -h n z) F (y|x) = exp ln F (y|x) ln F (y|x -h n z) ln F (y|x) -1 , it comes that F (t n |x -uh n ) F (t n |x) -1 ≤ |exp [Ch n ln t n ] -1| .

53

Applying Taylor, we get:

sup u∈Ω F (t n |x -uh n ) F (t n |x) -1 = O(h n ln t n ).
and therefore, in view of (2.5.2),

J2 = g (x) F (t n |x)O(h n ln t n ) Ω K(u) g(x -h n u) g (x) du = g (x) F (t n |x)O(h n ln t n )(1 + o(1)) = g (x) F (t n |x)O(h n ln t n ).
Then, we get

E K hn (x -X n ) 1 {Yn>tn} = g (x) F (t n |x) (1 + O(h n ln t n )) .
We state now the following technical lemma, which is proved in Mokkadem et al. (2009a), and which will be used throughout the demonstrations. Lemma 2.5.1.

Let (v n ) ∈ GS (v * ), (γ n ) ∈ GS (-α) and m > 0 such that m -v * ε > 0 where ε is dened in (4.2.1), and Π n in (2.2.3). Then, lim n→∞ v n Π m n n k=1 Π -m k γ k v k = 1 m -v * ε .
Moreover, for all positive sequences (α n ) such that lim n→∞ α n = 0, and all C ∈ R,

lim n→∞ v n Π m n n k=1 Π -m k γ k v k α k + C = 0. 2.5.2 Proof of Theorem 2.2.1
1. The application of Lemma 2.5.2, ensures that

E(a n (x)) = Π n n k=1 Π -1 k γ k mk (x) = Π n n k=1 Π -1 k γ k m k (x)g (x) (1 + O(h k ln t n )) .
In the case p ∈ (0, α/ (d + 2)], we have lim n→∞ nγ n > p; the application of lemma 2.5.1 ensures that

E(a n (x)) = a(x) + O(h n ln t n ),
and (2.2.10) follows. In the case p ∈ (α/ (d + 2) , 1/d), we have

h n ln t n = o γ n h -d n , Lemma 2.5.1 ensures E(a n (x)) -a(x) = O γ n h -d n
, which gives (2.2.11).

54

2. Now, we have

Var(a n (x)) = Π 2 n n k=1 Π -2 k γ 2 k E h -2d k K 2 x -X k h k [ln Y k -ln t n ] 2 1 {Y k >tn} -E 2 h -d k K x -X k h k [ln Y k -ln t n ] 1 {Y k >tn} .
Following the same steps of the proof of Theorem 1 in [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF], we obtain

Var(a n (x)) = Π 2 n n k=1 Π -2 k γ 2 k 6 K 2 1 h d k γ 2 (x)F (t n |x)g (x) (1 + o(1)) = Π 2 n n k=1 Π -2 k γ k γ k h d k 6 K 2 1 γ 2 (x)F (t n |x)g (x) (1 + o(1)) .
In the case when p ∈ [α/ (d + 2) , 1/d), we have lim n→∞ nγ n > α-pd 2 , and the application of Lemma 2.5.1 ensures that

Var(a n (x)) = = Q 2 n n k=1 Q -2 k β 2 k E h -2d k K 2 x -X k h k 1 {Y k >tn} -E 2 h -d k K x -X k h k 1 {Y k >tn} = Q 2 n n k=1 Q -2 k β 2 k K 2 2 h d k E h -d k H x -X k h k 1 {Y k >tn} -E 2 h -d k K x -X k h k 1 {Y k >tn} , with H(•) =: K 2 (•) K 2 2
also satisfying assumption (C6). Using Lemma 2.5.3, we get

Var(b n (x)) = Q 2 n n k=1 Q -2 k β 2 k K 2 2 h d k g (x) F (t n |x) (1 + O(h k ln t n )) -g 2 (x)F 2 (t n |x) (1 + O(h k ln t n )) ,
then, we have

Var(b n (x)) = K 2 2 g (x) F (t n |x)Q 2 n n k=1 Q -2 k β 2 k h d k + K 2 2 g (x) F (t n |x)Q 2 n n k=1 Q -2 k β 2 k O ln t n h d-1 k -g 2 (x)F 2 (t n |x)Q 2 n n k=1 Q -2 k β 2 k -g 2 (x)F 2 (t n |x)Q 2 n n k=1 Q -2 k β 2 k O(h k ln t n ).
In the case when p ∈ [b/ (d + 2) , 1/d), we have lim

n→∞ nβ n > (b -pd)/2
, and the application of Lemma 2.5.1 gives

Var(b n (x)) = 1 2 -(b -pd)ε 1 K 2 2 g (x) F (t n |x) β n h d n + O β n ln t n h d-1 n - 1 2 -bε 1 g 2 (x)F 2 (t n |x)β n + O(ln t n β n h n ),
which proves (2.2.17). In the case when p ∈ (0, b/ (d + 2)), we have Let us rst note that, for x such that b n (x) = 0, we have .5.4) It follows from (2.5.3), that the asymptotic behavior of γ n (x) -γ(x) can be deduced from the one of D n (x). 2.5.3). Now it follows from (2.5.4) that

β n h -d n = o(h 2 n ln 2 t n ), Lemma 2.5.1 ensures that Var(b n (x)) = O(h 2 n ln 2 t n ),
γ n (x) -γ(x) = D n (x) b(x) b n (x) , (2.5.3) with D n (x) = 1 b(x) (a n (x) -a(x)) - γ(x) b(x) (b n (x) -b(x)). ( 2
Var(D n (x)) = 1 b 2 (x) Var(a n (x)) - 2γ(x) b 2 (x) Cov(a n (x), b n (x)) + γ 2 (x) b 2 (x)
Var(b n (x)).

(2.5.5) By using Lemma 2.5.1 and choosing the stepsize (γ n ) = (n -1 ), computations provide 2.5.6) with

Cov(a n (x), b n (x)) = 1 n Q n n k=1 Q -1 k β k (J 1 -J 2 J 3 ) , (
J 1 = E K 2 h k (x -X k ) [ln Y k -ln t n ]1 {Y k >tn} , J 2 = m n (x) and J 3 = E K h k (x -X k ) 1 {Y k >tn} .
Following similar steps as Lemma 2 in Goegebeur et al. ( 2014) and Lemma 2.5.2, we infer that

J 1 = m n (x)g (x) K 2 2 h d k (1 + O(h k ln t n )) ,
J 2 and J 3 are already calculated in Lemmas 2.5.2 and 2.5.3. Then, the combination of (2.5.4), (2.5.5), (2.2.13), (2.2.17) and (2.5.6), gives (2.2.21), and the combination of (2.5.4), (2.5.5), (2.2.12), (2.2.16) and (2.5.6), gives (2.2.20).

2.5.5

Proof of Theorem 2.2.4

Let us at rst assume that, if p α/(d + 2), then

F (t n |x) -1 γ -1 n h d n ( γ n (x) -E ( γ n (x))) D -→ N (0, Var(x)) .
(2.5.7)

In the case when p > α/(d + 2), Part 1 of the theorem follows from the combination of (2.2.19) and (2.5.7). In the case when p = α/(d + 2), Parts 1 and 2 of the Theorem follow from the combination of (2.2.18) and (2.5.7). In the case p < b/(d + 2), (2.2.20) implies that

1 h n ln t n ( γ n (x) -E ( γ n (x))) P -→ 0,
and the application of (2.2.18) gives Part 2 of Theorem. Now (2.5.7) is proved. Relying on (2.5.4), we have

D n (x) -E[D n (x)] = 1 b(x) Π n n k=1 (Y k (x) -E[Y k (x)]) ,
where

Y k (x) = Π -1 k γ k Z k (x) -γ(x)η n η -1 k β k W k (x) , with Z n (x) = K hn (x -X n ) [ln Y n -ln t n ]1 {Yn>tn} , W n (x) = K hn (x -X n ) 1 {Yn>tn} and η n = Π -1 n Q n . Now, in the case when (β n ) = n -1 , we have η n = (nΠ n ) -1 and η -1 k β k = Π k . Then, Y k (x) = Π -1 k γ k Z k (x) -γ(x)(nΠ n ) -1 W k (x). Set T k (x) = Y k (x) -E [Y k (x)] .
(2.5.8)

Moreover, we have

s 2 n = n k=1 Var (T k (x)) = n k=1 Π -2 k γ 2 k Var (Z k (x)) + γ 2 (x)(nΠ n ) -2 n k=1 Var (W k (x)) -2γ(x)(nΠ n ) -1 n k=1 Π -1 k γ k Cov (Z k (x), W k (x)) := Γ 1 + Γ 2 + Γ 3 .
In addition, classical computations and applications of Lemma 2.5.1 ensure that

Γ 1 = Π -2 n γ 2 (x) 6 2 -(α -pd)ε K 2 1 g (x) F (t n |x) γ n h d n + o γ n h d n , Γ 2 = Π -2 n γ 2 (x) 1 1 + pd K 2 2 g (x) F (t n |x) nh d n + o 1 nh d n , Γ 3 = Π -2 n γ 2 (x) 2 1 + pdε K 2 2 g (x) F (t n |x) nh d n + o 1 nh d n .
As a matter of fact, we infer that

s 2 n = Π -2 n b 2 (x)F (t n |x) γ n h d n [Var(x) + o(1)] .
On the other side, we have, for all q > 0,

E | Y k (x) | 2+q = O 1 h (1+q)d k ,
and, since lim

n→∞ (nγ n ) > (α -pd) /2, there exists q > 0 such that lim n→∞ nγ n > 1+q 2+q (α -pd). Applying Lemma 2.5.1, we get n k=1 E | T k (x) | 2+q = O n k=1 Π -2-q k γ 2+q k E | Y k (x) | 2+q = O γ 1+q n Π 2+q n h (q+1)d n ,
and we thus obtain

1 s 2+q n n k=1 E | T k (x) | 2+q = 1 s 2(1+q/2) n O γ 1+q n Π 2+q n h (q+1)d n = O γ q 2 n h -dq 2 n = o (1) .
The convergence in (2.5.7) then follows from the application of Lyapounov's Theorem.

Introduction

Non parametric distribution function methods have a central position in statistics, and an enormous literature exists in this subject. Non parametric kernel type methods have been widely used in estimating distribution function. We can list without to trying to be exhaustive [START_REF] Nadaraya | Some New Estimates for Distribution Functions[END_REF], [START_REF] Azzalini | A note on the estimation of a distribution function and quantiles by a kernel method[END_REF], [START_REF] Reiss | Nonparametric estimation of smooth distribution functions[END_REF], [START_REF] Sarda | Smoothing parameter selection for smooth distribution functions[END_REF], [START_REF] Bowman | Bandwidth selection for the smoothing of distribution functions[END_REF] and [START_REF] Slaoui | The stochastic approximation method for the estimation of a distribution function[END_REF]. This estimation has been widely applied in many disciplines such as economics, nances, medicine, biology and various other situations.

In the current chapter, we consider the case of non parametric estimation of the distri-bution function, using a recursive kernel estimator version. In this way, the estimator can be updated with each additional new observation. This recursive scheme oers many advantages to recursive estimators: they do not require extensive storage of data and they are fast to compute.

In particular cases, they also appear as more ecient than classical estimators. Let X 1 , • • • , X n a sequence of random variables satisfy the α-mixing dependency property (see Denition 1.2.1) having a common unknown distribution function F with associated density f . Let us introduce a kernel function K dened on R (that is, a function satisfying R K(x)dx = 1), a function K (dened by K(z) = z -∞ K(u)du) and a bandwidth (h n ) (that is, a sequence of positive real number tending to zero when n goes to ∞). Let us recall that the usual kernel estimate of F (x) is given by

F n (x) = 1 n n i=1 K x-X i hn .
Whereas to construct a stochastic algorithm, which approximates the function F at a given point x, we dene an algorithm of search of the zero of the function φ : y → F (x) -y. We thus proceed in the following way:

(i) We set F 0 (x) ∈ [0, 1]; (ii) For all n 1, we set F n (x) = F n-1 (x) + γ n T n (x)
, where the stepsize (γ n ) is a positive sequence of real numbers decreasing to zero and T n is an observation of the function φ at the point F n-1 (x). Now, to dene T n (x), we follow the approach of [START_REF] Révész | Robbins-Monro procedure in a Hilbert space and its application in the theory of learning processes I[END_REF][START_REF] Révész | How to apply the method of stochastic approximation in the nonparametric estimation of a regression function[END_REF], [START_REF] Tsybakov | Recurrent estimation of the mode of a multidimensional distribution[END_REF] and more recently Slaoui (2014a,b) and we set

T n (x) = K h -1 n [x -X n ] -F n-1 (x).
Then, the estimator F n to recursively estimate the distribution function F at the point x can thus be written as

F n (x) = (1 -γ n )F n-1 (x) + γ n K x -X n h n . (3.1.1)
This estimator was proposed by [START_REF] Slaoui | The stochastic approximation method for the estimation of a distribution function[END_REF] in the case of independent data and whose large and moderate deviation principles was obtained by [START_REF] Slaoui | Data-driven deconvolution recursive kernel density estimators dened by stochastic approximation method[END_REF]. Moreover, we consider for simplicity that F 0 (x) = 0 and Π n = n j=1

(1 -γ j ). We can also consider the problem of bias reduction (see for instance the recent work of Slaoui (2018b), this would go well beyond the scope of the present chapter. By iteration, the estimator F n dened by (3.1.1) can be rewritten as

F n (x) = Π n n k=1 Π -1 k γ k K x -X k h k . (3.1.2)
Moreover, in the case when the stepsize (γ n ) is chosen equal to (n -1 ), the estimator F n dened by (3.1.2) can be rewritten as .1.3) This estimator was considered by [START_REF] Isogai | Nonparametric Recursive Kernel Estimators of a Distribution Function[END_REF]. The choice of such stepsize belongs to the subclass of the recursive kernel estimators of density, which have a minimum MSE (Mean Squared Error) or MISE (Mean Integrated Squared Error) (see Mokkadem et al. (2009a)). The aim of this chapter is to study the properties of the recursive distribution function estimator (3.1.3) in the case α-mixing data and its comparison with the kernel distribution estimator dened as

F n (x) = 1 n n k=1 K x -X k h k . ( 3
F n (x) = 1 n n i=1 K x -X i h n . (3.1.4)
A number of authors have studied the properties of the estimator (3.1.4) (see [START_REF] Nadaraya | Some New Estimates for Distribution Functions[END_REF], [START_REF] Reiss | Nonparametric estimation of smooth distribution functions[END_REF] and [START_REF] Hill | Kernel Estimation of a Distribution Function[END_REF]).

In recent years, data ows have become increasingly important in the eld of research. In this situation, the data arrives so fast that it is impossible to store it in a traditional database. In such a situation, the construction of a recursive estimator that does not require the storage of all the data in memory and that can be easily updated to process the online data is of great interest. This recursive estimator shows good theoretical properties, from the point of view of Mean Weighted Integrated Squared Error (MWISE) and almost sure convergence.

The purpose of this chapter is to generalize the recursive estimators proposed by [START_REF] Slaoui | The stochastic approximation method for the estimation of a distribution function[END_REF] to the case of α-mixing data. We rst compute the bias and the variance of the estimator F n dened by (3.1.3). It turns out that they heavily depend on the choice of the stepsize (γ n ). We show that using an adequate choice of the bandwidth (h n ), the expansion of the MWISE of the proposed estimator F n will be smaller than that of Nadaraya's estimator (3.1.4). We show also that estimator (3.1.3) can be very competitive to the estimator (3.1.4) in terms of estimation error and much better in terms of computational costs, especially for large n. This chapter is organized into ve sections. We study asymptotic properties in Section 2, while Section 3 is devoted to our application results. We conclude the chapter in Section 4, whereas the technical details are deferred to Section 5.

Assumptions and Main Results

Throughout this chapter, we consider stepsizes and bandwidths belonging to the following class of regularly varying sequences.

Denition 3.2.1. Let u ∈ R and (u n ) n 1 be a nonrandom positive sequence. We say that

u n ∈ GS(u) if lim n→∞ n 1 - u n-1 u n = u.
This condition was introduced by [START_REF] Galambos | Regularly Varying Sequences[END_REF] and by [START_REF] Mokkadem | A Companion for the KieferWolfowitzBlum Stochastic Approximation Algorithm[END_REF] in the context of stochastic approximation algorithms. Note that the acronym GS stand for (Galambos and Seneta). Typical sequences in GS(u) are, for b ∈ R, n u (log n) b , n u (log log n) b and so on.

The assumptions to which we shall refer are the following:

(H1) K : R -→ R is a continuous bounded function satisfying R K(x)dx = 1, R xK(x)dx = 0 and R x 2 K(x)dx < ∞. ( H2
) (i) The stepsize γ n ∈ GS(-α) with α ∈ (1/2, 1].
(ii) h n ∈ GS(-a) with a ∈ (0, 1).

(iii) lim n→∞ nγ n ∈ (min (2a, (α + a) /2) , ∞].
(H3) The density f is bounded, dierentiable and f is bounded.

(H4) The stepsize (γ n ) is a decreasing sequence and γ n -→ 0 as n -→ ∞.

(H5) The mixing coecient of the sequence (X i ) is geometry-dependent and satises α(n) = O(n -ρ ) for some ρ > 3.

(H6) There exist integer sequences (p n ) n and (q n ) n going to ∞ along with n such that

q n /p n -→ 0, q n h n -→ 0, p n h n -→ ∞ as n -→ ∞.
Moreover, for w := w n := n pn+qn (where [.] is the integer of the formula), we have

w n q -ρ n -→ 0 as n -→ ∞; w n γ n -→ 0, wγ -1 n Π 2 n (p n + q n ) ρ -→ 0, as n -→ ∞. (H7) n k=1 θ k θn < ∞, 1 k n θ k θn 2 < ∞, 1 i<j n θ i θ j θ 2 n < ∞ and 1 k n θ k θn 2 h k < ∞ where θ 2 k = Π -2 k γ 2 k
. Moreover, we use the following notations:

ξ = lim n→∞ (nγ n ) -1 , (3.2.1)
Z n (x) = K x -X n h n , (3.2.2)
µ 2 (K) = R z 2 K(z)dz, φ(K) = 2 R zK(z)K(z)dz, C 1 = R f 2 (x)dx, C 2 = R f (x) 2 f (x)dx, V F = R F (x)(1 -F (x))f (x)dx.
Our rst result are given in the following propositions, which give the bias and the variance of

F n respectively.
The following corollary is an immediate consequence of the previous remark. Now, we explicit the choices of (h n ) which minimize the M W ISE of our proposed recursive estimator dened by (3.1.3). Corollary 3.2.1. Let Assumptions (H1)-(H5) hold. To minimize the M W ISE of F n , the stepsize (γ n ) must be chosen in GS (-1), lim

n→∞ nγ n = γ 0 , the bandwidth (h n ) must equal (h n ) n = 2 -1/3 γ 0 - 2 3 1/3 C 1 φ(K) C 2 µ 2 2 (K) 1/3 n -1/3 , (3.2.6)
and then the corresponding M W ISE is equal to

M W ISE[F n ] = n -1 V F γ 2 0 2γ 0 -1 - 3 4 1 2 4/3 γ 2 0 (γ 0 -2/3) 2/3 C 4/3 1 φ(K) C 1/3 2 V F n -1/3 + o(n -1/3 ) .

Practical bandwidth selection

In order to give more details on the practical implementation of the proposed algorithm, we give rst a data driven bandwidth selection procedure to estimate the optimal bandwidth (3.2.6), we must estimate C 1 and C 2 . We followed the approach proposed in Slaoui (2014a,b), and we use the following kernel estimator of C 1 and C 2 respectively :

C 1 = Π n n n i,k=1 Π -1 k γ k b -1 k K b X i -X k b k , (3.3.1)
C 2 = Π 2 n n n i,j,k=1 Π -1 j Π -1 k γ j γ k b -2 j b -2 k K (1) b X i -X j b k K (1) b X i -X k b k , (3.3.2)
where K b is a kernel and b n is the associated bandwidth, K

is the rst derivative of a kernel K b and b n the associated bandwidth. It was shown in Slaoui (2014a,b) that in order to minimize the M ISE (Mean Integrated Squared Error) of C 1 (resp. of C 2 ), b n (resp. b n ) should belongs to GS (-2/5) (resp. GS (-3/10)). In practice, we take

b n = n -β min s, Q 3 -Q 1 1.349 , β ∈ (0, 1) (3.3.3)
(see [START_REF] Silverman | Density estimation for statistics and data analysis[END_REF]) with s the sample standard deviation, and Q 1 , Q 3 denoting the rst and third quartiles, respectively, here we take β = 2/5 to estimate b n and β = 3/10 to estimate b n . Moreover, in order to make a choice of the stepsize (γ n ), we choose γ 0 = 2/3 + ε such that ε is close to zero to ensure that the MISE of the recursive estimator F n is smaller than the one of the non recursive estimator F n (see [START_REF] Slaoui | The stochastic approximation method for the estimation of a distribution function[END_REF]).

We state the following theorem, which gives the weak convergence rate of our estimator (3.1.3). Theorem 3.3.1. Let Assumptions (H1)-(H6) hold.

If there exists

d 0 such that γ -1 n h 3 n -→ d, then γ -1 n (F n (x) -F (x)) D -→ N d 1/2 2(1 -2aξ) f (x)µ 2 (K), 1 2 -αξ F (x)(1 -F (x)) . (3.3.4) 2. If γ -1 n h 3 n -→ ∞, then 1 h 2 n (F n (x) -F (x)) P -→ 1 2(1 -2aξ) f (x)µ 2 (K).
Remark 3.3.1. The asymptotic bias and the asymptotic variance of the considered estimator F n in the case of of dependent data (α-mixing) are exactly the same as in the case of independent data, and consequently the convergence rate of F n in the two cases are the same, the main dierence between the two cases are certainly linked to the considered assumptions, the ones used in the case of dependent data are much more stronger than the ones used in the case of independent data. Moreover, the data-driven bandwidth procedures are the same in the two cases.

Let us underline that, when the bandwidth (h n ) n is chosen such that lim n→+∞ γ -1 n h 3 n = 0 (which corresponds to undersmoothing) and using the stepsize (γ n ) n = γ 0 n -1 n , we infer from (3.3.4), that the considered estimator F n fulls the following central limit theorem

√ n (F n (x) -F (x)) D -→ N 0, γ 2 0 2γ 0 -1 F (x)(1 -F (x)) .
We let φ denote the standard normal distribution function N (0, 1), and t α/2 be such that φ t α/2 = 1 -α/2 (where α ∈ (0, 1)). Then, the asymptotic condence interval of F (x) with level 1 -α, is given by

F n (x) ± t α/2 C (γ 0 ) F (x) (1 -F (x)) n ,
where, C (x) = x 2 (2x -1) -1 , this function reaching its minimum at the point x = 1. Then the best choice in point of view of estimation by condence intervals is obtained by considering the stepsize (γ n ) n = n -1 n , using this choice, the estimator F n fulls the central limit theorem

n 1/2 (F n (x) -F (x)) D -→ N (0, F (x)(1 -F (x))) .
It comes that, the asymptotic condence interval of F (x) with level 1 -α, in this special choice is given by

F n (x) ± t α/2 F (x) (1 -F (x)) n .
Remark 3.3.2. We can observe that in the special case when the bandwidth (h n ) n is chosen such that lim n→+∞ γ -1 n h 3 n = 0, and the stepsize (γ n ) n is chosen to be equal to n -1 n , the estimator F n fullls the same limit theorem as the one obtained for the empirical distribution, and consequently the two estimators (F n and the empirical distribution) have the same asymptotic condence interval of F . Theorem 3.3.2 (Uniform convergence). Let assumptions (H1) -(H3) hold, F is uniformly continuous and there exists η > 0 such that z -→ z η |F (x)| is a bounded function. We let C be a compact set of R. Then, we have

sup x∈C |F n (x) -F (x)| = o (1) a.s. as n -→ ∞.

Simulation study

The aim of our applications is to compare the performance of Nadaraya's estimator dened in (3.1.4) with the proposed recursive kernel distribution function estimator under α-mixing condition (1.2.1), dened in (3.1.3), using the Plug-in method of bandwidth selection.

3.4.1

The study design

We consider the following simulation design, we simulate N = 500 samples of sizes, n = 50, n = 100, n = 150 and a sequence of m-dependent variables

X i = i+m j |Z j |,
where (Z j ) j are generated from the following mixture distribution:

Z ∼ 1 2 N (2.5, 6) + 1 2 N (9, 1) .
In order to calculate the MWISE of the proposed recursive kernel distribution function F n , we need to use the following quantities:

• The Fonction K, we use the normal kernel.

• The stepsize (γ n ) = (γ 0 n -1 ), where γ 0 = 2/3 + c, with c ∈ [0, 1/3].

• The bandwidth (h n ) n , we consider the plug-in method, given in [START_REF] Slaoui | The stochastic approximation method for the estimation of a distribution function[END_REF], that there is chosen to be equal to Moreover, some numerical results of φ(K) and µ 2 (K) are given for some standard kernels (see Table 1 in [START_REF] Slaoui | The stochastic approximation method for the estimation of a distribution function[END_REF]).

  2 -1/3 γ 0 - 2 3 1/3 C 1 φ(K) C 2 µ 2 2 (K) 1/3 n -1/3   , Y ∼ 1 2 N (2.
In order to calculate the MWISE of the non recursive kernel distribution function F n , we need to use these quantities:

• The Fonction K, we use the normal kernel.

• The bandwidth (h n ), we consider the plug-in method (see [START_REF] Slaoui | The stochastic approximation method for the estimation of a distribution function[END_REF]).

Results

For each conguration of the simulation design parameters, we calculate the M W ISE of the non recursive estimator (3.1.4) and the recursive estimator (3.1.3). From Table 1, the proposed recursive estimator of F n (x) outperformed the non recursive estimator F (x) in all the considered situations. We can observe that the MWISE decrease as m increase. We can observe also that the MWISE decrease as the sample size n increase.

Conclusion

This chapter proposes an automatic bandwidth selection of the recursive density estimators under α-mixing condition (1.2.1). The proposed estimators asymptotically follow normal distribution.

The proposed estimators are compared to the non recursive distribution function estimator under α-mixing condition. We showed that using a specic plug-in bandwidth selection method and some particularly stepsizes, the proposed recursive estimators can give better results compared to the non recursive distribution function estimator under α-mixing condition in terms of estimation error. However, the main advantage of the recursive method is considerably faster than the classical one; see, for instance, Mokkadem et al. (2009a) and Slaoui (2014a) in the framework of density of probability estimation, [START_REF] Slaoui | The stochastic approximation method for the estimation of a distribution function[END_REF] in the framework of distribution estimation, [START_REF] Slaoui | Plug-in bandwidth selector for recursive kernel regression estimators dened by stochastic approximation method[END_REF]Slaoui ( , 2016a) ) in the framework of regression estimation and [START_REF] Slaoui | On the choice of smoothing parameters for semi-recursive nonparametric hazard estimators[END_REF] in the framework of hazard function. In conclusion, the proposed recursive estimators allowed us to obtain quite better results compared to the non recursive density estimator under α-mixing condition in terms of estimation error and much better in terms of computational costs.

Proofs

Before giving the outlines of the proofs, we introduce the following technical lemma, which is proved in Mokkadem et al. (2009a), and which will be used throughout the demonstrations.

Lemma 3.6.1. Let (v n ) ∈ GS (v * ), (γ n ) ∈ GS (-α) and m > 0 such that m -v * ε > 0 where ε is dened in (4.2.1), then

lim n→∞ v n Π m n n k=1 Π -m k γ k v k = 1 m -v * ε .
Moreover, for all positive sequence (α n ) such that lim n→+∞ α n = 0, and all C ∈ R, In view of (3.1.1) and (3.2.2), we have

lim n→∞ v n Π m n n k=1 Π -m k γ k v k α k + C = 0.
F n (x) -F (x) = (1 -γ n )F n-1 (x) + γ n Z n (x) -F (x) = (1 -γ n )(F n-1 (x) -F (x)) + γ n [Z n (x) -F (x)] = n j=1 (1 -γ j )(F 0 (x) -F (x)) + n-1 i=1 n j=i+1 (1 -γ j )γ i [Z i (x) -F (x)] +γ n [Z n (x) -F (x)] = Π n n k=1 Π -1 k γ k (Z k (x) -F (x)) + Π n (F 0 (x) -F (x)).
It implies that

E[F n (x)] -F (x) = Π n n k=1 Π -1 k γ k (E[Z k (x)] -F (x)) + Π n (F 0 (x) -F (x)).
Then, an integration by parts ensures that

E[Z k (x)] = R K x -t h k f (t)dt = R K(z)F (x + h k z) dz. (3.6.1) It follows that E[Z k (x)] -F (x) = R K(z) [F (x + h k z) -F (x)] dz = h 2 k 2 f (x)µ 2 (K) + β k (x) with β k (x) = R K(z) F (x + h k z) -F (x) -zh k f (x) - 1 2 z 2 h 2 k f (x) dz,
and, since f is bounded and continuous, we have lim k→∞ β k (x) = 0.

In the case a > α/3, we have

h 2 n = o √ γ n h n and lim n→∞ (nγ n ) > (a + α)/2, then Lemma 4.5.1 ensures that E[F n (x)] -F (x) = Π n n k=1 Π -1 k γ k o γ k h k + O(Π n ) = o γ n h n . (3.6.2)
Moreover, in the case when a α/3, we have lim n→∞ (nγ n ) > 2a; the application of Lemma 4.5.1 ensures that .6.3) Then, the combination of (3.6.2) and (3.6.3) give (3.2.3). First, we have

E[F n (x)] -F (x) = 1 2 f (x)µ 2 (K)Π n n k=1 Π -1 k γ k [h 2 k + o(1)] + Π n (F 0 (x) -F (x)) = 1 2(1 -2aξ) f (x)µ 2 (K)[h 2 n + o(1)]. ( 3
Var [F n (x)] = Var Π n n k=1 Π -1 k γ k Z k (x) = Π 2 n n k=1 Π -2 k γ 2 k Var (Z k (x)) + 2Π 2 n 1 i<j n Π -1 i γ i Π -1 j γ j Cov (Z i (x), Z j (x)) =: I 1 + I 2 ,
where

I 1 = Π 2 n n k=1 Π -2 k γ 2 k Var (Z k (x)) , I 2 = 2Π 2 n 1 i<j n Π -1 i γ i Π -1 j γ j Cov (Z i (x), Z j (x)) .
Now, in order to compute I 1 , we use the following decomposition .6.4) An integration by parts ensures that

I 1 = Π 2 n n k=1 Π -2 k γ 2 k E Z 2 k (x) -E 2 (Z k (x)) . ( 3
E Z 2 k (x) = R K 2 x -t h k f (t)dt = 2 R K (-z) K(z)F (x + zh k )dz = v k (x) + F (x) -h k f (x)φ(K) (3.6.5) with v k (x) = 2 R K(z)K(-z) [F (x + zh k ) -F (x) -zh k f (x)] dz.
Moreover, it follows from (3.6.1), that

E (Z k (x)) = F (x) + v k (x), (3.6.6) with v k (x) = R K(z) [F (x + zh k ) -F (x)] dz.
Then, the combination of (3.6.4), (3.6.5) and (3.6.6) gives In the case a α/3, we have lim n→∞ nγ n > (a + α)/2, and the application of Lemma 4.5.1 gives

I 1 = Π 2 n n k=1 Π -2 k γ 2 k F (x) -h k f (x)φ(K) + v k (x) -F 2 (x) -v 2 k (x) -2F (x) v k (x) = F (x) (1 -F (x)) Π 2 n n k=1 Π -2 k γ 2 k -f (x)φ(K)Π 2 n n k=1 Π -2 k γ 2 k h k + v k (x) -v 2 k (x) -2F (x) v k (x) Π 2 n n k=1 Π -2 k γ 2 k . ( 3
I 1 = γ n 2 -αξ F (x)(1 -F (x)) - γ n h n 2 -(a + α)ξ f (x)φ(K) + o(γ n h n ). ( 3
I 1 = γ n 2 -αξ F (x)(1 -F (x)) + Π 2 n n k=1 Π -2 k γ k o(h 4 k ) = γ n 2 -αξ F (x)(1 -F (x)) + o(γ n ).
(3.6.9)

Now, in the case when a ∈ (0, α/4), we have γ n = o(h 4 n ), and lim n→∞ nγ n > 2a, then the application of Lemma 4.5.1 gives

I 1 = Π 2 n n k=1 Π -2 k γ k o(h 4 k ) = o(h 4 n ).
(3.6.10)

Then, (3.2.4) follows from the combination of (3.6.8), (3.6.9) and (3.6.10). Let us now compute I 2 , we have

I 2 = 2Π 2 n 1 i<j n Π -1 i γ i Π -1 j γ j Cov (Z i (x), Z j (x)) = 2Π 2 n 1 i<j n B i,j , s where B i,j = Π -1 i γ i Π -1 j γ j Cov K x-X i h i , K x-X j h j . |B i,j | = Π -1 i γ i Π -1 j γ j Cov K x -X i h i , K x -X j h j = Π -1 i γ i Π -1 j γ j R 2 K x -t 1 h i K x -t 2 h j f X i ,X j (t 1 , t 2 )dt 1 dt 2 - R K x -t 1 h i f (t 1 )dt 1 R K x -t 2 h j f (t 2 )dt 2 .
Assumption (H4), a simple change of variables and an integration by parts imply .6.11) Next, to evaluate the asymptotic behavior of I 2 , we dene the sets

|B i,j | M Π -1 i γ i Π -1 j γ j R 2 K (s 1 ) K (s 2 ) ds 1 ds 2 = O(Π -1 i γ i Π -1 j γ j ). ( 3
F 1 = {(i, j) such that 1 |i -j| σ n } ,
and

F 2 = {(i, j) such that σ n + 1 |i -j| n -1}
where

σ n = o(n). Let J 1,n = 2Π 2 n (i,j)∈F 1 B i,j and J 2,n = 2Π 2 n (i,j)∈F 2 B i,j .
Then it follows from (3.6.11) and (H4), that

|J 1,n | 2M Π 2 n (i,j)∈F 1 Π -1 i γ i Π -1 j γ j
and applying Lemma 4.5.1, we infer that

|J 1,n | 2M σ n γ n 1 2 -αξ = O σ n γ n 2 -αξ .
In order to compute J 2,n , we use the Davydov inequality for mixing processes (see [START_REF] Rio | Théorie asymptotique des processus aléatoires faiblement dépendants. Mathématiqueset applications[END_REF], p. 10, Formula 1.12a). This leads, for all i = j, to

Cov Π -1 i γ i Z i (x), Π -1 j γ j Z j (x) cα (|i -j|) .
Therefore, using (H5), we have

J 2,n 2cΠ 2 n n j=1 σn+1<k n-1 α (k) < 2cnΠ 2 n n-1 σn+1 k -v dk = O nΠ 2 n σ 1-v n . By choosing σ n = n 1/v γ -1/v n Π 2/v n 1 2-αξ -1/v
, the assumption (H5), ensures that

I 2 = J 1,n + J 2,n = O n 1/v γ (v-1)/v n Π 2/v n (2 -αξ) (v-1)/v = o(1),
which conclude the proof. We have

γ -1 n (F n (x) -F (x)) = γ -1 n (F n (x) -E (F n (x))) + γ -1 n (E (F n (x)) -F (x)) = Υ 1,n + Υ 2,n .
(3.6.12)

First, we determine that Υ 2,n are negligible, whereas Υ 1,n is asymptotically normal.

In order to establish the asymptotic normality, dealing with strong mixing random variables (under (H5)), we use the well-known sectioning device introduced by (Doob (1955), p. 228-232). We rst split the sum in (3.6.15) below into large p n blocks and small q n blocks under (H6). For that, observe that w n (p n + q n ) n and w n (p n + q n )/n -→ 1 as n -→ ∞ and, for

j = 1, • • • , w n , partition the set {1, 2, • • • , n} into (2w n + 1
) subsets with w =: w n blocks of size p n and k n blocks of size q n , as follows: Let 3.6.13) where .6.14) In order to prove (3.6.14), we set

y mn = km+p-1 i=km L i (x), y mn = lm+q-1 j=lm L j (x), y wn = n k=w(p+q)+1 L k (x) (
k m = (m -1)(p + q) + 1, l m = (m -1)(p + q) + p + 1, m = 1, • • • , w. Let us rst assume that if a α/3, we have γ -1 n (F n (x) -F (x)) D → N 0, 1 2 -αξ F (x)(1 -F (x)) . ( 3
L k (x) = Π -1 k γ k (Z k (x) -E(Z k (x))
), then, it comes from (3.1.1), that .6.15) Moreover, since we have

F n (x) -E (F n (x)) = Π n n k=1 Π -1 k γ k (Z k (x) -E(Z k (x)) = Π n n k=1 L k (x). ( 3
Υ 1,n (x) = Π n γ -1 n n k=1 L k (x).
We infer that, 3.6.16) and then we prove that .6.17) To proof (3.6.16), we use Tchebychev's inequality. Then, we need to show that

Υ 1,n (x) = Π n γ -1 n w m=1 y mn + w m=1 y mn + y wn := Π n γ -1 n {T n,1 + T n,2 + T n,3 } . We let ϕ n = Π n γ -1 n . Let us rst show that ϕ n (T n,2 + T n,3 ) P -→ 0, as n -→ ∞, (
ϕ n T n,1 D -→ N 0, 1 2 -αξ F (x)(1 -F (x)) . ( 3
ϕ 2 n E T 2 n,2 + T 2 n,3 -→ 0, n -→ ∞.
For the rst term, we consider the following decomposition

ϕ 2 n E T 2 n,2 = ϕ 2 n w m=1 lm+q-1 i=lm E L 2 i (x) + 2ϕ 2 n 1 i<j w E y in y jn +2ϕ 2 n w m=1 lm i<j lm+q+1 Cov (L i (x)L j (x)) = Θ 1 + Θ 2 + Θ 3 ,
where

Θ 1 = ϕ 2 n w m=1 lm+q-1 i=lm E L 2 i (x) Θ 2 = 2ϕ 2 n 1 i<j w E y in y jn Θ 3 = 2ϕ 2 n w m=1 lm i<j lm+q+1
Cov (L i (x)L j (x)) .

The combination of (3.6.8), (3.6.11) and (3.6.13), together with (H6), (H7), ensure that

Θ 1 -→ 0 and Θ 3 -→ 0 as n -→ ∞.
Now, in order to compute Θ 2 , we use [START_REF] Rio | Théorie asymptotique des processus aléatoires faiblement dépendants. Mathématiqueset applications[END_REF], p.10, Formula 1.12a, to infer that

E y i y j Cα (q n + (j -i -1)(p n + q n )) ,
we then deduce that

Θ 2 Cγ -1 n Π 2 n w-1 i=1 w-i-1 j=0 (q n + (j -i -1)(p n + q n )) -ρ Cγ -1 n Π 2 n (p n + q n ) -ρ w-1 i=1 ∞ l=0 l -ρ O wγ -1 n Π 2 n (q n + q n ) ρ = o(1).
Now, in order to proof (3.6.17), we show that 3.6.19) where

E [exp (itα n T n,1 )] - Wn m=1 E [exp (itα n y mn )] -→ 0 (3.6.18) and α 2 n Wn m=1 E y 2 mn 1 {ym>ξα -1 n σ(x)} -→ 0 as n -→ ∞, (
σ 2 (x) = 1 2-αξ F (x)(1 -F (x)).
Using [START_REF] Volkonskii | Some limit theorems for random functions[END_REF] inequality, we obtain

E exp itα n wn m=1 y mn - Wn m=1 E [exp (itα n y mn )] 16(w n -1)α(q n ) -→ 0 as n -→ ∞
which, under (H6), yields to (3.6.18).

For (3.6.19), the combination of (3.6.13) together with (H1), ( H3) and (H6), for n large enough, ensure that the set y mn , n > ξα -1 σ(x) become empty which completes the proof (see [START_REF] Khardani | Recursive Kernel Density Estimation and Optimal Bandwidth Selection Under α: Mixing Data[END_REF]).

3.6.4

Proof of Theorem 3.3.2

First, using the compactness property of the set C, we use the fact that, for some

(x k ) 1≤k≤γn , C ⊂ γn k=1 B (x k , a n ), where γ n ∼ a -1 n with a n = h 1 α +1 n .
Moreover, for any x ∈ C, we set k (x) = arg min k x k -x . We infer that, for any x ∈ C, we have

sup x∈C |F n (x) -E [F n (x)]| ≤ sup x∈C F n (x) -F n x k + sup x∈C F n x k -E F n x k + sup x∈C E F n x k -E [F n (x)] =: T 1,n + T 2,n + T 3,n .
(3.6.20)

We let α denote the Hölder order of K and K H its corresponding Hölder norm. Then, it follows from (3.1.2) that for any x ∈ C

F n (x) -F n x k ≤ Π n n k=1 Π -1 k γ k K X k -x h k -K X k -x k h k ≤ 2 K H Π n n k=1 Π -1 k γ k x -x k h k α ≤ 2 K H Π n n k=1 Π -1 k γ k h k ,
we then get T 1,n = o (1) and T 3,n = o (1). Moreover, we set ρ > 0 and M such that

F ∞ z >M |K (z)| dz ≤ ρ/2. Since the application of Lemma 4.5.1 ensures that Π n n k=1 Π -1 k γ k = 1 + o (1)
, then, it follows from (3.6.1)

F n x k -E F n x k ≤ Π n n k=1 Π -1 k γ k E K X k -x k h k ≤ Π n n k=1 Π -1 k γ k R K (z) F x k + zh k dz ≤ ρ 2 + z ≤M |K (z)| F x k dz +Π n n k=1 Π -1 k γ k z >M |K (z)| F x k + zh k -F x k dz.
Then, the uniform continuity of F combined with the dominate convergence and the existence of η > 0 such that z -→ z η |F (x)| is a bounded function ensure that T 2,n = o (1). Then the combination of Proposition 3.2.1 and (3.6.20) concludes the proof of Theorem 3.3.2.

Introduction

The estimation of mode function stands for a classical problem in statistics which has whetted considerable interest in various elds of applications. Indeed, it is widely used in machine learning applications and, in particular, in clustering methods (see [START_REF] Cheng | Mean shift, mode seeking, and clustering[END_REF]; [START_REF] Sheikh | Mode-seeking by medoidshifts[END_REF]; Jiang and Kpotufe (2017)), computer vision (see [START_REF] Yin | Fast mode decision and motion estimation for JVT/H.264[END_REF]; [START_REF] Tao | Colour image segmentation based on mean shift and normalized cuts[END_REF]), power systems (see [START_REF] Williams | Mode estimation of model-based programs: monitoring systems with complex behavior[END_REF]; Nezam Sarmadi and Venkatasubramanian (2014)), control (see [START_REF] Hofbaur | Mode estimation of probabilistic hybrid systems[END_REF]) and bioinformatics (see [START_REF] Hedges | Comparison of mode estimation methods and application in molecular clock analysis[END_REF]). Multiple research works related to this topic within the frame work of non parametric estimation have been elaborated. Among the most prominent ones, we mention [START_REF] Parzen | On estimation of a probability density function and mode[END_REF], [START_REF] Samanta | Nonparametric estimation of the mode of a multivariate density[END_REF] and [START_REF] Tsybakov | Recurrent estimation of the mode of a multidimensional distribution[END_REF]. Recently, there has been a spate of interest in recursive estimation which has drawn the attention of multiple researchers. The basic merit of the recursive estimator lies in the fact that it can not only be updated with each additional new observation especially in large sample sizes but it can also be much better in terms of computational costs. In this work, our central focus is upon a recursive kernel estimator of the mode function dened by stochastic approximation method. Let X 1 , • • • , X n be identically distributed R d -valued random vectors satisfy the α-mixing dependency property (see Denition 1.2.1) and let f denote the probability density of X

i , i = 1, • • • , n.
We consider a compact set Ω such that Ω ⊂ R d , and we dene the mode as follows

θ := arg max y∈Ω f (y).
We assume that θ is unique.

In order to dene our estimator of the mode, we rst begin by constructing a stochastic algorithm for the estimation of the function f at a point x. We present an algorithm to search for the zero of the function g : y -→ f (x) -y. Following Robbins-Monro's procedure, this algorithm is dened below as (i) f 0 (x) is an arbitrary choice belonging to R,

(ii) ∀n 1, we set f n (x) = f n-1 (x) + γ n W n (x)
, where the stepsize (γ n ) n is a sequence of positive real numbers that goes to zero and W n (x) is an observation of the function g at the point f n-1 (x).

To construct W n (x), we follow the approach of [START_REF] Révész | Robbins-Monro procedure in a Hilbert space and its application in the theory of learning processes I[END_REF][START_REF] Révész | How to apply the method of stochastic approximation in the nonparametric estimation of a regression function[END_REF] and [START_REF] Tsybakov | Recurrent estimation of the mode of a multidimensional distribution[END_REF] which are based on the classical property of stochastic algorithms (E W n (x)|F n-1 1 = 0, where F n-1 1 stands for the σ-eld of events generated by {X 1 , • • • , X n-1 }). In addition, we introduce a kernel K (which is a function satisfying R d K(z)dz = 1), and a bandwidth (h n ) (which is a sequence of positive real numbers that goes to zero when n -→ ∞), and we set W

n (x) = K hn (x -X n ) -f n-1 (x), with K h (x) := h -d K h -1
x . Therefore, the recursive estimator of the density function f at the point x can be written as

f n (x) = π n f 0 + π n n k=1 π -1 k γ k h -d k K x -X k h k with π n = n k=1
(1 -γ k ). Our estimator of mode θ is dened as the random variable θ n maximizing the recursive estimator f n of f , which is expressed as

θ n := arg max t∈Ω f n (t). (4.1.1)
The mode estimator has been investigated by several authors. Based on independent and identically distributed (iid) random data, the weak consistency and the asymptotic normality (A4) The joint density f (i,j) of (X i , X j ) exists for all (i, j), and there exists a constant M > 0 such that

sup |i-j| 1 sup t 1 ,t 2 ∈Ω f (i,j) (t 1 , t 2 ) -f (t 1 )f (t 2 ) < M.
(A5) The mixing coecient of the X i 's satises α(n) = O(n -ν ) for some ν 3. (A6) The mode θ satises the following property: for any ε > 0 and x, there exists η = 0 such that

|θ -x| > ε implies that |f (θ) -f (x)| > η. (A7) (i) n 1/ν γ 1-1/ν n -→ n-→∞ 0. (ii) a(dν -2) -α(d + 2) > 6 if a ≥ α/(d + 4) a(d -2ν -6) -α > 6 if a < α/(d + 4).
Remark 4.2.1. Assumption (A1) on the kernel is widely used in the recursive and non recursive framework for the functional estimation. Assumptions (A2) on the stepsize and the bandwidth are used in the recursive framework for the estimation of the density function (Mokkadem et al. (2009a); [START_REF] Slaoui | Large and moderate principles for recursive kernel density estimators dened by stochastic approximation method[END_REF]Slaoui ( , 2014aSlaoui ( , 2018b))). Hypothesis (A2)(i) and (A2)(ii) ensure that the bandwidth (h n ) and the stepsize (γ n ) go to zero as n goes to innity. Moreover, the stepsize (γ n ) goes to zero more rapidly than the bandwidth (h n ). Assumption (A2)(iii) on the limit as n goes to innity of (nγ n ) is usual in the framework of stochastic approximation algorithms. It implies that the limit of (nγ n ) -1 is nite. Assumption (A3) on the function f allows us to calculate the properties of our estimator. Condition (A4) is needed to calculate the covariance. (A5) states a condition on the mixing coecient. Assumption (A6) is classical in mode estimation. Finally, hypothesis (A7) provides a condition for the bandwidth allowing the estimation of the covariance term.

Throughout this chapter, we shall use the following notation:

ε = lim n→∞ (nγ n ) -1 , (4.2.1) µ 2 j = R d z 2 j K(z)dz, ∀j ∈ {1, • • • , d} , (4.2.2) f (2) ij (x) = ∂ 2 f (x) ∂x i ∂x j , Z n (x) = 1 h d n K x -X n h n .
The almost sure convergence is denoted by a.s..

• Y ∼ 1 2 N (2.5, 6) + 1 2 N (9, 1) .

• Y ∼ 1 2 N (2, 6) + 1 2 N (8, 1) .

Next, we calculate the ISE (Integrated Squared Error) and the IAE (Integrated Absolute Error) of the two estimators;

ISE = 1 N N i=1 θ [i] n -θ 2 and IAE = 1 N N i=1 θ [i] n -θ ,
where θ

[i]

n corresponds to the mode estimator computed from the ith sample. In order to calculate the ISE and the IAE of the two mode estimators, we need to use the following quantities:

• The normal kernel function K.

• The stepsize (γ n ) n = (n -1 ) n .

• The bandwidth (h n ) n is chosen with plug-in method, given in Slaoui (2014a).

Results

For each conguration of the simulation design parameters, we calculate the ISE and the IAE of the non recursive estimator (4.3.1) and the recursive estimator (4.1.1). From Table 1, Table 2, Table 3 and Table 4, it is clear that, the proposed recursive estimator (4.1.1) outperformed the non recursive estimator (4.3.1) in all the considered situations. We can observe that the ISE decreases as m increases. We can observe also that the ISE decreases as the sample size n increases. This simulation study shows the good performance of the recursive estimator with an appropriate choice of stepsize and bandwidth parameters.

Conclusion

In this chapter, we attempted to elaborate a recursive kernel mode estimator based on stochastic approximation algorithm. We established the strong consistency of this estimator under α-mixing condition. Investing the same selected parameters in Mokkadem et al. (2009a), which minimize the mean squared error of recursive density estimator, the proposed recursive mode estimator maintains the same convergence rate with non recursive mode estimator dened by (4.3.1).

The two previous estimators are asymptotically equivalent. In addition, the main merit of our estimator resides in its update, when a new sample information becomes available. Tackling this area is extremely interesting as it oers new perspectives for future works to consider multiple directions within this framework. This involves the elaboration of recursive mode estimation for dependent strong mixing functional data like in [START_REF] Slaoui | Recursive nonparametric regression estimation for independent functional data[END_REF]. Furthermore, our proposed

For more details about previous Lemma 4.5.3, we refer to Rio (2000), p. 87, 6.19b. Lemma 4.5.4. Under Assumptions (A1)-(A7), we have Now, we set for t ∈ Ω with

T i (t) = π -1 i γ i h -d i K t -X i h i -E K t -X i h i . ( 4
Ti (t) = π -1 i γ i h -d i K t -X i h i -K t * k -X i h i -π -1 i γ i h -d i E K t -X i h i -E K t * k -X i h i .
As a matter of fact, we have In order to investigate U 1 , we observe that

π n n i=1 Ti (t) π n n i=1 π -1 i γ i h -d i K t -X i h i -K t * k -X i h i +π n n i=1 π -1 i γ i h -d i E K t -X i h i -K t * k -X i h i := V 1 (t) + V 2 (t).
Assumptions (A1), (4.5.3) and ( 4.5.4) and the application of Lemma 4.5.1 provide

V 1 (t) cπ n n i=1 π -1 i γ i h -d i t -t * k h i cπ n n i=1 π -1 i γ i h -(d+1) i |t -t * k | cb n h -(d+1) n 1 1 + a(d + 1)ε cγ 1/2 n h -d/2 n 1 1 + a(d + 1)ε = O γ n h -d n , and
V 2 (t) cπ n n i=1 π -1 i γ i h -(d+1) i E [|t -t * k |] = O γ n h -d n .
Thus, we get

U 1 = O γ n h -d n a.s as n -→ ∞.
Now, in order to study U 2 , we use Lemma 4.5.3. For that, let Now, from (4.5.6) as well as under assumptions (A1) and (A4), we have

W i = π n T i (t * k ) = π n π -1 i γ i h -d i K t * k -X i h n -E K t * k -X i h n . ( 4
|Cov (W i , W j )| = E π 2 n π -1 i π -1 j γ i γ j h -d i h -d j K t * k -X i h n K t * k -X j h n -E π n π -1 i γ i h -d i K t * k -X i h n E π n π -1 j γ j h -d j K t * k -X j h n = π 2 n π -1 i π -1 j γ i γ j h -d i h -d j E K t * k -X i h n K t * k -X j h n -E K t * k -X i h n E K t * k -X j h n = π 2 n π -1 i π -1 j γ i γ j R 2d K(t 1 )K(t 2 ) f (i,j) (t * k -t 1 h i , t * k -t 2 h j ) -f (t * k -t 1 h i )f (t * k -t 2 h j )| dt 1 dt 2 M π 2 n π -1 i γ i π 1 j γ j = O(π 2
n π -1 i γ i π 1 j γ j ). (4.5.7)

Next, to asses the term S 2 * n , we use a technique developed by [START_REF] Masry | Recursive probability density estimation for weakly dependent process[END_REF]. We dene the sets Applying the upper bound in (4.5.7), we have

F 1,n M π 2 n i,j∈F 1 π -1 i γ i π -1 j γ j
F 1,n M β n γ n 1 2 -αε = O (β n γ n ) .
For F 2 , we use the Davydov inequality for mixing processes (see Rio 2000, p. 10, Formula 1.12a). This leads us to get, for all i = j |Cov (W i , W j )| cα (|i -j|) .

Therefore, using (A5), we obtain

f (θ n ) -f (θ) = (θ n -θ)f (θ) + 1 2 (θ n -θ) 2 f (2) (θ * n ) = 1 2 (θ n -θ) 2 f (2) (θ * n ),
where θ * n is between θ and θ n . Therefore, based on (4.5.11) and (A3), we get

|θ n -θ| 2 |f (θ n ) -f (θ)| f (2) (θ * n ) 2 sup t∈Ω |f n (t) -f (t)| | f (2) (θ * n ) | .
Thus, by (4.2.1) the proof holds.

vantage of our estimator resides in its update, when a new sample information becomes available.

At this stage of synthesis, it is noteworthy that our thesis would be valuable in terms of opening further fruitful lines of investigation and oering promising future perspectives. Indeed, this thesis may be extended in several ways:

• First, we may extend our recursive extreme value index estimator to the case of censored data. We can also propose a new estimator of the conditional extreme quantile using our recursive estimator dened by (2.2.7) and compare it to the classical Weissman estimator.

• Second, we may equally apply Bernstein and Lagrange polynomials to estimate extreme value index and extreme quantile functions (See [START_REF] Slaoui | Recursive density estimators based on Robbins-Monro's scheme and using Bernstein polynomials[END_REF] and [START_REF] Helali | Estimation of a distribution function using Lagrange polynomials with TchebychevGauss points[END_REF]).

• Third we may explore of a recursive mode estimation for dependent strong mixing functional data like in Slaoui (2019[START_REF] Slaoui | Recursive nonparametric regression estimation for independent functional data[END_REF] and for dependent strong mixing spatial data like in Bouzebda andSlaoui (2018, 2020). Furthermore, our proposed recursive kernel mode estimator is promising and can be extended in such a way as addressing recursive non parametric estimation in the Bayesian work (see [START_REF] Boukabour | Semiparametric Bayesian networks for continuous data[END_REF]).

2. 1

 1 Pattern of γ(•) on [0, 1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Boxplots of the N = 500 estimates of our ve considered estimators (N R,

  In this chapter, we are basically interested in heavy tails. More precisely, we assume that the conditional survival function of Y given X = x satises (C1): F (y|x) = y -1 γ(x) l(y|x), where γ(•) is an unknown positive continuous function of the covariate x called the tail function and for a xed x, l(•|x) is a function that varies slowly at innity, i.e for all λ > 0,lim y→∞ l(λy|x) l(y|x) = 1. Condition (C1) means that the conditional distribution of Y given X = x isin the Frechet maximum domain of attraction. The tail function γ(x) is the conditional extreme value index function which needs to be adequately estimated from the available data. (C2): l(•|x) is normalized. The Karamata representation (Theorem 1.3.1 given in Bingham et al. (1987)) of the slowly-varying function, l(•|x), can be written as l(y|x) = c(x) exp y 1 ε(z|x) z dz , where c(•) is a positive function and ε(z|x) -→ 0 as z -→ ∞. Thus, l(•|x) is dierentiable and the function ε(•|x) is given by ε(z|x) = z l (z|x) l(z|x) . (C3): There exists a strictly negative function ρ(•), a strictly positive function γ(•) and a rate function b(•|x), b(y|x) -→ 0 as y -→ ∞, of constant sign for large values of y such that for all υ > 0 lim y→∞ F (υy|x)

Corollary 2 . 2 . 2 .

 222 Let the assumptions of Theorem 2.2.2 hold. To minimize the asymptotic variance of the estimator b n , b must be chosen equal to 1, (β n ) n must satisfy lim n→∞ nβ n = 1 -pd, and we then have

Figure 2 . 1 :

 21 Figure 2.1: Pattern of γ(•) on [0, 1]

Figure 2 . 2 :

 22 Figure 2.2: Boxplots of the N = 500 estimates of our ve considered estimators (N R, R1, R2, R3, R4) in points x = 0.1, 0.2, 0.3 (1st line), x = 0.4, 0.5, 0.6 (2nd line) and x = 0.7, 0.8, 0.9 (3rd line), and dashed lines represent the values of reference estimator γ(•) in each point x as mentioned above.

  .6.7) Since f , f are continuous and bounded, we have lim k→∞ v k (x) = 0 and lim k→∞ v k (x) = 0.

  . as n -→ ∞.Proof ofLemma 4.5.4. The proof relies upon the following assertion: the compact setΩ can be covered by a nite number λ n of balls B k (t * k , b n ) centered at t * k , 1 k λ n where b n satises b n = γ 1/2 n h 1+d/2 n . (4.5.3)Since Ω is bounded, one can nd l > 0 such that λ n ≤ lb -1 n . For any t ∈ Ω, there exists k such that|t -t * k | b n .(4.5.4)

|

  Cov (W i , W j ) | = i =j |Cov (W i , W j )| +Mokkadem et al. (2009a) for more details about computation of the variance.

F 1 =F 2 =

 12 {(i, j) : 1 |i -j| β n } and {(i, j) : β n + 1 |i -j| n -1} where β n = o(n). Let F 1,n = i,j∈F 1 |Cov (W i , W j )| and F 2,n = i,j∈F 2 |Cov (W i , W j )| .

  γ j+1 ) • • • (1 -γ j+k )

F 2 .

 2 Choosing β n = nγ -1 n 1/ν and under (A7)(i), we obtainS 2 * n = F 1,n + F 2,n = O n 1/ν γ 1-1

:

  of fact, we apply Lemma 4.5.3 in the case a ≥ α/(d + 4). We obtain, for any k = c (Γ 1,n + Γ 2,n ) .

  sup t∈Ω |f n (t) -f (t)| + |f n (θ n ) -f (θ)| . (4.5.10) Since |f n (θ n ) -f (θ)| = sup t∈Ω f n (t) -sup t∈Ω f (t) sup t∈Ω |f n (t) -f (t)| , then we have |f (θ n ) -f (θ)| 2 sup t∈Ω |f n (t) -f (t)| . (4.5.11) The a.s. consistency of θ n follows then immediately from (4.2.1) and (A6). Now a taylor expansion provides

,

  Departing from Table2.2 and Figure2.3, we infer that the IAE and the ISE of the proposed recursive estimator are smaller than those of the non recursive estimator set forward by[START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF]. Thus, demonstrating the eectiveness of our considered estimator.

		0.8	recursive estimator
			non-recursive estimator
		0.7
		0.6
	Index function	0.5
		0.4
		0.3
		0.2
		0.0	0.2	0.4	0.6	0.8	1.0
				Observation
	Figure 2.3: Qualitative comparaison between the non recursive estimator (2.1.1) and the
	proposed recursive one (2.2.7).

  which gives(2.2.16).

	2.5.4	Proof of Theorem 2.2.3

  Then, (2.2.18) follows from (2.2.10), (2.2.14) and (2.5.3) whereas (2.2.19) follows from (2.2.11), (2.2.15) and (

Table 3 .

 3 1: MWISE (approximated using N = 500 trials) of the non recursive estimator and the recursive estimator.

			5, 6) + 1 2 N (9, 1)
		n = 50	n = 100	n = 150
		M W ISE	M W ISE	M W ISE
	m = 2
	Non recursive 0.003061	0.001641	0.001123
	Recursive	0.002391	0.001453	0.001040
	m = 4
	Non recursive 0.002877	0.001565	0.001087
	Recursive	0.002272	0.001396	0.001016
	m = 6
	Non recursive 0.002740	0.001510	0.001030
	Recursive	0.002176	0.001357	0.000972
	m = 8
	Non recursive 0.002666	0.001462	0.001025
	Recursive	0.002146	0.001319	0.000968
	with C 1 and C 2 are given in Slaoui (2014b) (see respectively equations (3.3.1) and (3.3.2)).

  .6.8) Moreover, in the case when a ∈ [α/4, α/3), we have γ n h n = o(h 4 n ), and lim

	n→∞	nγ n > α/2, Lemma
	4.5.1 then ensures that

-(α -pd)ε K 2

Acknowledgements

Countless people supported my eort on this thesis. At rst, I pray god to give me

which proves (2.2.13). In the case when p ∈ (0, α/ (d + 2)), we have γ n h -d n = o(h 2 n ln 2 t n ), Lemma 2.5.1 ensures that Var(a n (x)) = o(h 2 n ln 2 t n ), which yields (2.2.12).

2.5.3

Proof of Theorem 2.2.2

1. First, the application of Lemma 2.5.3 provides

Now, in the case when p ∈ (0, b/ (d + 2)], we have lim n→∞ nβ n > p; the application of Lemma 2.5.1 ensures that

and (2.2.14) follows. In the case when p ∈ (b/ (d + 2) , 1/d), we have

, which gives (2.2.15).

Now, we have

Var(b n (x))

Proposition 3.2.1. (Bias of F n ) Let assumptions (H1) -(H3) hold, and assume that f is continuous at x. Then

Moreover, in order to measure the quality of our proposed estimator F n dened in (3.1.3), we consider the Mean Weighted Integrated Squared Error (MWISE):

The following proposition gives the MWISE of the estimator F n for the mixing case. Proposition 3.2.3. Let Assumptions (H1)-(H6) hold, and assume that f is continuous at x.

The following remark follows immediately from the previous proposition.

Remark 3.2.1. One can infer from (3.2.3) and (3.2.4), that in the special case a = α/3, we have an asymptotic expression of the bias and the variance of F n , the same remark can be done from (3.2.5), then, under the assumptions of the proposition 3.2.3, we have in the case when

Then it comes that we can obtain an optimal bandwidth in the special case a = α/3, which will be very helpful for practice.

of the kernel sample mode was addressed by [START_REF] Parzen | On estimation of a probability density function and mode[END_REF]. This result was extended in several directions by [START_REF] Cherno | Estimation of the mode[END_REF], [START_REF] Eddy | Optimum kernel estimators of the mode[END_REF][START_REF] Eddy | The asymptotic distributions of kernel estimators of the mode[END_REF] and [START_REF] Vieu | A note on density mode function[END_REF]. Strong consistency was explored by [START_REF] Nadaraya | On nonparametric estimates of density functions and regression curves[END_REF] and [START_REF] Van Ryzin | On strong consistency of density estimates[END_REF]. Asymptotic normality of kernel estimate of the mode was elaborated by [START_REF] Romano | On weak convergence and optimality of kernel density estimates of the mode[END_REF]. The multidimensional study of the mode was carried out by [START_REF] Samanta | Nonparametric estimation of the mode of a multivariate density[END_REF] and [START_REF] Konakov | On the asymptotic normality of the mode of multidimensional distributions[END_REF].

Based on dependent random data, some studies have been performed for mode estimation. In φmixing condition as well as the conditional case, the strong consistency was enacted by Collombs et al (1987). In alpha mixing case, the strong consistency was established by Ould [START_REF] Ould Saïd | Estimation non paramétrique du mode conditionnel application à la prévision[END_REF] and the asymptotic normality was set forward by [START_REF] Louani | Asymptotic normality of kernel estimators of the conditional mode under strong mixing hypothesis[END_REF]. Numerous works were conducted, under censored and truncated data, to explore the property of non parametric mode estimators (see [START_REF] Louani | On the asymptotic normality of the kernel estimators of the density function and its derivatives under censoring[END_REF][START_REF] Ould Saïd | Strong uniform consistency of nonparametric estimation of the censored conditional mode function[END_REF] and [START_REF] Gannoun | Asymptotic properties of kernel estimators of the mode under censoring[END_REF]).

The majority of properties of mode estimators are related to those of density estimators. We need always to handle the density case before that of the mode. This chapter investigates the estimation of the mode, which is based on non parametric recursive kernel density estimator developed by Mokkadem et al. (2009a), under strong mixing conditions. The rest of the chapter is organized as follows. In Section 4.2, the assumptions and main results are displayed. Section 4.3 is devoted to simulation study. Finally, a conclusion is presented in Section 4.4. The details of proofs are exhibited in Section 4.5 along with some auxiliary results.

Assumptions and main results

We consider stepsizes and bandwidths, which belong to the following class of regularly varying sequences.

Denition 4.2.1. Let γ ∈ R and (γ n ) n 1 be a nonrandom positive sequence. We state that

The assumptions to which we shall refer are the following:

(A1) The kernel function K:R d -→ R is a bounded probability density, lipschitz and satises for all j ∈ {1, . . . , d}, R z j K(z)dz j = 0 and

(A3) f is bounded, twice dierentiable on Ω, and, for all i, j ∈ {1,

Proposition 4.2.2. Under the assumption of Proposition 4.2.1, we have

a.s. as n -→ ∞.

Simulation study

In this section, we aim to compare our proposed recursive kernel estimator of mode, dened by

where (Y j ) j are generated from the following mixture distributions: recursive kernel mode estimator is promising and can be extended in such a way as addressing recursive non parametric estimation in the Bayesian work (see [START_REF] Boukabour | Semiparametric Bayesian networks for continuous data[END_REF]).

Proofs

Before setting the outlines of the proofs, we introduce the following technical lemma, which is proved in Mokkadem et al. (2009a), and which will be used throughout the demonstrations.

Lemma 4.5.1.

Then,

Moreover, for all positive sequence (α n ) n such that lim n→∞ α n = 0, and all C ∈ R,

Proof of Proposition 4.2.1. The proof rests on the following decomposition

and is based on the proofs of the following three lemmas.

Lemma 4.5.2. Under Assumptions (A1)-(A3), we have

as n -→ ∞.

The proof of Lemma 4.5.2 is presented in Mokkadem et al. (2009a). Let (W i ) i∈N be a sequence of centered real random variables, with a strong mixing coecient α

Hence, for all ε > 0 and r > 1, there exists a constant c such that

By taking

n log n and r = c log n(log 2 n) 1/ν (4.5.9) and using Taylor series expansion of log(1 + x) as well as (4.5.8a)-( 4.5.9), we infer

where log 2 n = log(log n) for n > 2. Consequently,

Now, referring to (A7)(ii), we have

corresponding to the general term of the convergent Bertrand series. For Γ1,n , an appropriate choice of ε 0 can be made O n -3/2 , which corresponds to the general term of convergent series. Hence, n 1 Γ1,n + Γ2,n < ∞, and therefore (4.5.2a) follows by applying Borel Cantelli Lemma. The same steps shall be used in the second case if a < α/(d + 4). The result (4.5.2b) is a consequence of Borel Cantelli Lemma after applying Lemma 4.5.3 and choosing

Chapter 5

Conclusion and perspective

For clarity and methodological reasons, the basic concepts and properties used in the subsequent analysis are highlighted in chapter 1.

In chapter 2, we tackled the estimation of the conditional extreme value index γ(x) of a heavy-tailed distribution when some random covariate information is available. We elaborated recursive kernel estimator of the extreme value index function based on the stochastic approximation algorithm. We demonstrated that using some particular stepsizes and a specic bandwidth selection through a cross-validation procedure, our recursive estimator could be very competitive to Hill's non recursive version in terms of estimation error and computational costs. We illustrated this performance via simulations and real data.

In chapter 3, we extended the work of [START_REF] Slaoui | The stochastic approximation method for the estimation of a distribution function[END_REF] to the case of α-mixing data. We established the central limit theorem and the uniform convergence for the proposed estimator under some mild conditions. We conrmed that using a specic plug-in bandwidth selection method and some particularly stepsizes, the proposed recursive estimator yielded better results compared to Nadaraya's non recursive distribution estimator under α-mixing condition. However, the basic merit of the recursive method resides essentially in the fact that it is much faster than the classical one. Eventually, these theoretical results were corroborated through a few simulations.

In chapter 4, we elaborated a recursive kernel mode estimator based on stochastic approximation algorithm. We established the strong consistency of this estimator under α-mixing condition. Investing the same selected parameters in Mokkadem et al. (2009a), which minimize the mean squared error of recursive density estimator, the proposed recursive mode estimator maintains the same convergence rate with non recursive mode estimator dened by (4.3.1). We shown that two previous estimators are asymptotically equivalent. In addition, the main ad-