Mikael Escobar-Bach

Yuri Goegebeur

Armelle Guillou

Local estimation of the conditional stable tail dependence function

Keywords: Conditional stable tail dependence function, empirical process, stochastic convergence

We consider the local estimation of the stable tail dependence function when a random covariate is observed together with the variables of main interest. Our estimator is a weighted version of the empirical estimator adapted to the covariate framework. We provide the main asymptotic properties of our estimator, when properly normalized, in particular the convergence of the empirical process towards a tight centered Gaussian process. The finite sample performance of our estimator is illustrated on a small simulation study and on a dataset of air pollution measurements.

Introduction

A central topic in multivariate extreme value statistics is the estimation of the extremal dependence between two or more random variables. [START_REF] Ledford | Modelling dependence within joint tail regions[END_REF] introduced the coefficient of tail dependence as a summary measure of extremal dependence, and also proposed an estimator for this parameter. See also [START_REF] Peng | Estimation of the coefficient of tail dependence in bivariate extremes[END_REF], [START_REF] Beirlant | Some comments on the estimation of a dependence index in bivariate extreme value in statistics[END_REF], [START_REF] Beirlant | Bias-reduced estimators for bivariate tail modelling[END_REF], [START_REF] Goegebeur | Asymptotically unbiased estimation of the coefficient of tail dependence[END_REF], [START_REF] Dutang | Robust and bias-corrected estimation of the coefficient of tail dependence[END_REF] for alternative estimators of this parameter. Other examples of summary dependence measures for extremes can be found in [START_REF] Coles | Dependence measures for extreme value analyses[END_REF]. As an alternative to these summary measures, one can work with functions that give a complete characterisation of the extremal dependence, like the spectral distribution function [START_REF] Einmahl | Estimating the spectral measure of an extreme value distribution[END_REF], the Pickands dependence function [START_REF] Pickands | Multivariate extreme value distributions[END_REF] or the stable tail dependence function [START_REF] Huang | Statistics of bivariate extremes[END_REF]. These functions can be seen as the analogues of copulas in classical statistics. In the present paper we focus on the stable tail dependence function.

For any arbitrary dimension d, let pY p1q , . . . , Y pdq q be a multivariate random vector with continuous marginal distribution functions F 1 , . . . , F d . The stable tail dependence function is defined for each y i R , i 1, . . . , d, as lim tÑV tP ¡ 1 ¡ F 1 pY p1q q ¤ t ¡1 y 1 or . . . or 1 ¡ F d pY pdq q ¤ t ¡1 y d © Lpy 1 , . . . , y d q,

provided that this limit exists, which can be rewritten as lim tÑV t 1 ¡ F F ¡1 1 p1 ¡ t ¡1 y 1 q, . . . , F ¡1 d p1 ¡ t ¡1 y d q ¨$ Lpy 1 , . . . , y d q, where F is the multivariate distribution function of the vector pY p1q , . . . , Y pdq q. Now, consider a random sample of size n drawn from F and an intermediate sequence k k n , i.e. k Ñ V as n Ñ V with k{n Ñ 0. Let us denote y py 1 , . . . , y d q a vector of the positive quadrant R d and Y pjq k,n the k¡th order statistic among n realisations of the margins Y pjq , j 1, . . . , d.

The empirical estimator of L is then given by

p L k pyq 1 k n i1 1l tY p1q i ¥Y p1q n¡rky 1 s 1,n or ... or Y pdq i ¥Y pdq n¡rky d s 1,n u .
The asymptotic behaviour of this estimator was first studied by [START_REF] Huang | Statistics of bivariate extremes[END_REF]; see also Drees alternative estimators for L were introduced. In the present paper we extend the empirical estimator to the situation where we observe a random covariate X together with the variables of main interest pY p1q , . . . , Y pdq q. We consider thus a regression problem where we want to describe the extremal dependence between the random variables pY p1q , . . . , Y pdq q given some observed value x for the covariate X. Our approach is nonparametric and based on local estimation in the covariate space.

In the univariate context there is a quite extensive literature on estimation of tail parameters in presence of random covariates. In the framework of heavy-tailed distributions, nonparametric kernel methods were introduced by Daouia et al. (2011), who used a fixed number of extreme conditional quantile estimators to estimate the conditional extreme value index, for instance using the Hill [START_REF] Hill | A simple general approach to inference about the tail of a distribution[END_REF] and Pickands [START_REF] Pickands | Statistical inference using extreme order statistics[END_REF] estimators, whereas Goegebeur et al. where an adjustment of the moment estimator, originally proposed by [START_REF] Dekkers | A moment estimator for the index of an extreme-value distribution[END_REF], to this setting of local estimation has been proposed. On the contrary, the development of extreme value methodology for regression problems with a multivariate response vector is still in its infancy. In de Carvalho and Davison (2014), a procedure was introduced to infer about extremal dependence in the presence of qualitative independent variables, that is, an ANOVAtype setting. [START_REF] Portier | On the weak convergence of the empirical conditional copula under a simplifying assumption[END_REF] considered the estimation of a bivariate extreme value distribution under the simplifying assumption that the dependence between Y p1q and Y p2q does not depend on the value taken by the covariate, so that the dependence of the model on the covariates is only through the marginal distributions. Escobar-Bach et al. (2017a) studied the robust estimation of the conditional Pickands dependence function using the minimum density power divergence criterion, adapted to the context of local estimation. However, in that paper it is assumed that a random sample from a conditional bivariate extreme value distribution is available. In the present paper we relax this assumption and introduce a local estimator for the conditional stable tail dependence function assuming only that we have data available from a distribution with a dependence structure converging to that of an extreme value distribution.

Thus, we extend the above framework to the case where the vector pY p1q , . . . , Y pdq q is recorded along with a random covariate X R p . In that context, the stable tail dependence function together with the marginal distribution functions depend on the covariate X. In the sequel, for We establish the weak convergence of the empirical process of the properly normalized estimator using Donsker results for changing function classes and arguments based on the theory of Vapnik-Červonenkis classes (VC -classes). To the best of our knowledge this type of regression problem has not been considered in the multivariate extreme value literature.

The remainder of the paper is organised as follows. In the next section we introduce the local estimator for the conditional stable tail dependence function and study its asymptotic properties. In first instance we assume that the marginal conditional distribution functions are known, whereafter this assumption is removed and the unknown marginal conditional distribution functions are estimated locally using a kernel method. Finally, in Section 3, we illustrate the finite sample behaviour of our estimator with a small simulation study and on a dataset of air pollution measurements. All the proofs of the results are collected in the Appendix.

Estimator and asymptotic properties

Denote pY, Xq : pY p1q , . . . , Y pdq , Xq, a random vector satisfying (1), and let pY 1 , X 1 q, . . . , pY n , X n q, be independent copies of pY, Xq, where X R p has density function f . As is usual in the extreme value context, we consider an intermediate sequence k k n , i.e. k Ñ V as n Ñ V with k{n Ñ 0. Let us denote y : py 1 , . . . , y d q a vector of the positive quadrant R d . The event A t,y is defined for any t ¥ 0 and y R d as A t,y :

3 1 ¡ F 1 pY p1q |X q ¤ t ¡1 y 1 or . . . or 1 ¡ F d pY pdq |X q ¤ t ¡1 y d A ,
and A piq t,y denotes its analogue for observation pY i , X i q, i 1, . . . , n. The conditional empirical estimator is then given for any x R p by p T k py|xq : 1

k n i1 K h px ¡ X i q 1l ! 1¡F 1 pY p1q i |X i q¤ k n y 1 or ... or 1¡F d pY pdq i |X i q¤ k n y d ) 1 k n i1 K h px ¡ X i q 1l A piq n{k,y , (2)
where K h p.q : Kp.{hq{h p with K a joint density function and h h n is a positive non-random sequence satisfying h n Ñ 0 as n Ñ V.

The aim of the paper is to derive stochastic convergence results for empirical processes based on

(2), with y r0, T s d , T ¡ 0, but with the covariate argument fixed, meaning that we will focus our study only around one reference position x 0 IntpS X q, the interior of the support S X of f .

In order to derive the asymptotic behaviour of p T k py|x 0 q, we need to introduce some conditions mentioned below and well-known in the extreme value framework. Let }.} be some norm on R p , and denote by B x prq the closed ball with respect to }.} centered at x and radius r ¡ 0. First order condition: The limit in (1) exists for all x S X and y R d , and the convergence is uniform on r0, T s d ¢ B x 0 prq for any T ¡ 0 and a r ¡ 0.

Second order condition: For any x S X there exist a positive function α x such that α x ptq Ñ 0 as t Ñ V and a non null function M x such that for all y R d lim tÑV 1 α x ptq ttP pA t,y |X x q ¡ Lpy|xqu M x pyq, uniformly on r0, T s d ¢ B x 0 prq for any T ¡ 0 and a r ¡ 0.

Due to the regression context, we need some Hölder-type conditions.

Assumption pDq. There exist M f ¡ 0 and η f ¡ 0 such that |fpxq ¡ f pzq| ¤ M f }x ¡ z} η f , for all px, zq S X ¢ S X . Assumption pLq. There exist M L ¡ 0 and η L ¡ 0 such that |Lpy|xq ¡ Lpy|zq| ¤ M L }x ¡ z} η L , for all px, zq B x 0 prq ¢ B x 0 prq, r ¡ 0, and y r0, T s d , T ¡ 0.

Assumption pAq. There exist M α ¡ 0 and η α ¡ 0 such that |α x ptq ¡ α z ptq| ¤ M α }x ¡ z} ηα , for all px, zq S X ¢ S X and t ¥ 0. Also a usual condition is assumed on the kernel function K.

Assumption pK 1 q. K is a bounded density function on R p with support S K included in the unit ball of R p with respect to the norm }.}.

Marginal conditional distributions known

In this section, we restrict our interest to the case where the marginal conditional distribution functions F j p.|xq, j 1, . . . , d, are known. We start by showing the convergence in probability of our main statistic under some weak assumptions.

Lemma 2.1 Let y R d . Assume the first order condition, (K 1 ) and that the functions f and x Ñ Lpy|xq are continuous at x 0 IntpS X q non-empty. If for n Ñ V we have k Ñ V and h Ñ 0 in such a way that k{n Ñ 0 and kh p Ñ V, then for x 0 such that f px 0 q ¡ 0, we have p T k py|x 0 q P ÝÑ f px 0 qLpy|x 0 q.

This result indicates that in order to estimate Lpy|x 0 q, the statistic p T k py|x 0 q will need to be divided by an estimator for f px 0 q. Our main objective in this section is to show the weak convergence of the stochastic process

5 c kh p £ p T k py|x 0 q p f n px 0 q ¡ Lpy|x 0 q ¡ α x 0 ¡ n k © M x 0 pyq , y r0, T s d C , (3)
for any T ¡ 0, where p f n is the usual kernel density function estimator for f :

p f n pxq : 1 n n i1 K h px ¡ X i q,
see e.g. [START_REF] Parzen | On estimation of a probability density function and mode[END_REF]. Note that for convenience we use here for p f n the same kernel function K and bandwidth parameter h as for p T k py|x 0 q.

As a preliminary step we deduce the covariance structure of the limiting process (apart from the scaling by 1{ p f n px 0 q). T k py|x 0 q, p T k py I |x 0 q © Ñ f px 0 q}K} 2 2 Lpy|x 0 q Lpy I |x 0 q ¡ Lpy y I |x 0 q ¨, as n Ñ V.

Here, y y I : py 1 y I 1 , y 2 y I 2 , . . . , y d y I d q and }K} 2 :

³ S K K 2 puqdu.
We derive now the weak convergence of (3) using Donsker type results for empirical processes with changing function classes and arguments based on the theory of V C-classes, as formulated in van der [START_REF] Van Der Vaart | Weak convergence and empirical processes, with applications to statistics[END_REF]. These allow us to obtain weak convergence results by mainly focusing on the class of functions involved in our estimator. It should be mentioned that our main weak convergence results are derived in the usual Skorohod space, here Dpr0, T s d q equipped with the sup norm }.} V .

Theorem 2.1 Assume the second order condition, pDq, pLq, pAq, pK 1 q, and px, yq Ñ M x pyq being continuous on B x 0 prq ¢ r0, T s d , with B x 0 prq S X . Consider sequences k Ñ V and h Ñ 0 as n Ñ V, in such a way that k{n Ñ 0, kh p Ñ V and c kh p h minpη f ,η L ,ηαq Ñ 0 and c kh p α x 0 pn{kq Ñ λ x 0 R .

Then, for x 0 such that f px 0 q ¡ 0, the process

5 c kh p £ p T k py|x 0 q p f n px 0 q ¡ Lpy|x 0 q ¡ α x 0 ¡ n k © M x 0 pyq , y r0, T s d C ,
weakly converges in Dpr0, T s d q towards a tight centered Gaussian process tB y , y r0, T s d u, for any T ¡ 0, with covariance structure given by Cov B y , B y I ¨ }K} 2 2 f px 0 q Lpy|x 0 q Lpy I |x 0 q ¡ Lpy y I |x 0 q ¨, where y, y I r0, T s d .

Marginal conditional distributions unknown

In this section, we consider the general framework where all F j p.|xq, j 1, . . . , d, are unknown conditional distribution functions. We want to mimic what has been done in the previous section in case where these conditional distributions are assumed to be known. To this aim, we consider the random vectors

¡ p F n,1 pY p1q i |X i q, p F n,2 pY p2q i |X i q, . . . , p F n,d pY pdq i |X i q, X i © , i 1, . . . , n,
for suitable estimators p F n,j of F j , j 1, . . . , d. Then, similarly as in Section 2.1, we study the

statistic q T k py|x 0 q : 1 k n i1 K h px 0 ¡ X i q 1l ! 1¡ p F n,1 pY p1q i |X i q¤ k n y 1 or ... or 1¡ p F n,d pY pdq i |X i q¤ k n y d ) .
Our final goal is still the same, that is the weak convergence of the stochastic process

5 c kh p £ q T k py|x 0 q p f n px 0 q ¡ Lpy|x 0 q ¡ α x 0 ¡ n k © M x 0 pyq , y r0, T s d C .
The idea will be to decompose the process

3 c kh p ¡ q T k ¡ Er q T k s © py|x 0 q, y r0, T s d A into the two terms 3 c kh p ¡ p T k ¡ Er p T k s © py|x 0 q c kh p ¡ r q T k ¡ p T k s ¡ Er q T k ¡ p T k s © py|x 0 q, y r0, T s d A . (4)
The first term in the above display can be dealt with using the results of Section 2.1 whereas we have to show that the second term is uniformly negligible. To achieve this objective, let us introduce the following empirical kernel estimator of the unknown conditional distribution functions p F n,j py|xq :

°n i1 K c px ¡ X i q1l tY pjq i ¤yu °n i1 K c px ¡ X i q , j 1, . . . , d,
where c : c n is a positive non-random sequence satisfying c n Ñ 0 as n Ñ V. Here we kept the same kernel K as for q

T k py|x 0 q, but of course any other kernel function can be used.

We need to impose again some assumptions, in particular a Hölder-type condition on each marginal conditional distribution function F j similar to the one imposed on the conditional stable tail dependence function.

Assumption pF m q. There exist M F j ¡ 0 and η F j ¡ 0 such that |F j py|xq ¡ F j py|zq| ¤ M F j }x ¡ z} η F j , for all y R, all px, zq S X ¢ S X and j 1, . . . , d.

Concerning the kernel K a stronger assumption than pK 1 q is needed. Assumption pK 2 q. K satisfies Assumption pK 1 q, there exists δ, m ¡ 0 such that B 0 pδq S K and Kpuq ¥ m for all u B 0 pδq, and K belongs to the linear span (the set of finite linear combinations) of functions k ¥ 0 satisfying the following property: the subgraph of k, tps, uq : kpsq ¥ uu, can be represented as a finite number of Boolean operations among sets of the form tps, uq : qps, uq ¥ ϕpuqu, where q is a polynomial on R p ¢R and ϕ is an arbitrary real function.

This assumption has already been used in [START_REF] Giné | Rates of strong uniform consistency for multivariate kernel density estimators[END_REF] and Escobar-Bach et al.

(2017a).

In particular, we refer to the latter to enunciate the following lemma that measures the discrepancy between the conditional distribution function F j and its empirical kernel version p F n,j .

Lemma 2.3 Assume that there exists b ¡ 0 such that f pxq ¥ b, dx S X R p , f is bounded, and pK 2 q and pF m q hold. Consider a sequence c tending to 0 as n Ñ V such that for some q ¡ 1 | log c| q nc p ÝÑ 0. Also assume that there exists an ε ¡ 0 such that for n sufficiently large

inf xS X λ ptu B 0 p1q : x ¡ cu S X uq ¡ ε, (5)
where λ denotes the Lebesgue measure. Then for any 0 η minpη F 1 , . . . , η F d q, we have

sup py,xqR¢S X § § § p F n,j py|xq ¡ F j py|xq § § § o P £ max £ | log c| q nc p , c η , for j 1, . . . , d.
This rate of convergence allows us to study the second term in (4) and to show that it is uniformly negligible.

Theorem 2.2 Assume that there exists b ¡ 0 such that f pxq ¥ b, dx S X R p , f is bounded, pK 2 q, pF m q, pDq, the first order condition and condition [START_REF] Capéraà | Bivariate distributions with given extreme value attractor[END_REF], and also for any y r0, T s d that

x Ñ Lpy|xq continuous at x 0 IntpS X q non-empty. Consider sequences k Ñ V, h Ñ 0 and c Ñ 0 as n Ñ V, such that k{n Ñ 0, kh p Ñ V, and with for some q ¡ 1 and

0 η minpη F 1 , . . . , η F d q n h p k r n : n h p k max £ | log c| q nc p , c η ÝÑ 0, as n Ñ V. Then sup yr0,T s d c kh p § § § q T k ¡ p T k ¡ E q T k ¡ p T k % § § § py|x 0 q o P p1q.
Finally, the decomposition (4) combined with Theorem 2.2 and the results from Section 2.1, yield the desired final result of this paper.

Theorem 2.3 Assume the second order condition, px, yq Ñ M x pyq continuous on B x 0 prq ¢ r0, T s d , with B x 0 prq S X , and that there exists b ¡ 0 with f pxq ¥ b, dx S X R p , f bounded.

Under pDq, pLq, pAq, pF m q, pK 2 q and condition (5

), consider sequences k Ñ V, h Ñ 0 and c Ñ 0 as n Ñ V, such that k{n Ñ 0, kh p Ñ V with c kh p h minpη f ,η L ,ηαq Ñ 0, c kh p α x 0 pn{kq Ñ λ x 0 R ,
and for some q ¡ 1 and

0 η minpη F 1 , . . . , η F d q n h p k max £ | log c| q nc p , c η ÝÑ 0.
Then, the process

5 c kh p £ q T k py|x 0 q p f n px 0 q ¡ Lpy|x 0 q ¡ α x 0 ¡ n k © M x 0 pyq , y r0, T s d C ,
weakly converges in Dpr0, T s d q towards a tight centered Gaussian process tB y , y r0, T s d u, for any T ¡ 0, with covariance structure given in Theorem 2.1.

3 Simulation and real data analysis

f Y 1 ,Y 2 py 1 , y 2 q c 1 ¡ θ 2 2π ¢ 1 y 2 1 ¡ 2θy 1 y 2 y 2 2 ν ¡ ν 2 2 , py 1 , y 2 q R 2 ,
and θ being the Pearson correlation coefficient. The stable tail dependence function can be described as

Lpy 1 , y 2 |θq y 2 F ν 1 £ py 2 {y 1 q 1{ν ¡ θ c 1 ¡ θ 2 c ν 1
y 1 F ν 1 £ py 1 {y 2 q 1{ν ¡ θ c 1 ¡ θ 2 c ν 1 ,
where F ν 1 is the distribution function of the univariate Student distribution with ν 1 degrees of freedom. Asymptotic independence can be reached for θ ¡1 and complete positive dependence for θ 1. We set θ X, where X is uniformly distributed on r0, 1s. This model satisfies the second order condition with

M x py 1 , y 2 q C y 2{ν 1 2 F ν 3 £ py 2 {y 1 q 1{ν ¡ θ c 1 ¡ θ 2 c ν 3 y 2{ν 1 1 F ν 3 £ py 1 {y 2 q 1{ν ¡ θ c 1 ¡ θ 2 c ν 3 ' , C : ¡ ν 2{ν 1 π 1{ν pν 1q 2pν 2q £ Γ ν 2 Γ ν 1 2 ¨2{ν , α x ptq t ¡2{ν .
Moreover, one could check that the uniform property in the first and second order conditions are verified since we have continuity of the involved functions. The model satisfies also conditions pDq, pLq, pAq and pF m q. In the simulations we set ν 1. Model 2: We consider a particular case of the Archimax bivariate copulas introduced in

Cpy 1 , y 2 |xq 2 1 Lpy ¡1 1 ¡ 1, y ¡1 2 ¡ 1|xq @ ¡1 ,
where we use for L the asymmetric logistic stable tail dependence function defined by

Lpy 1 , y 2 |xq p1 ¡ t 1 qy 1 p1 ¡ t 2 qy 2 pt 1 y 1 q θx pt 2 y 2 q θx % 1{θx
, where 0 ¤ t 1 , t 2 ¤ 1, and θ x minp1{x, 100q, with the covariate X uniformly distributed on r0, 1s. The marginal distributions are taken to be unit Fréchet. This model satisfies our second order condition with

M x py 1 , y 2 q y 2 1 f 1 Lpy 1 , y 2 |xq y 2 2 f 2 Lpy 1 , y 2 |xq ¡ L 2 py 1 , y 2 |xq, α x ptq t ¡1 ,
and also satisfies pDq, pLq, pAq and pF m q. In the simulations, different values for the pair pt 1 , t 2 q have been tried but the results seem to be not too much influenced by them, thus we exhibit only the results in case pt 1 , t 2 q p0.4, 0.6q which corresponds to an asymmetric tail dependence function.

To compute our estimator s L k , two sequences

c j : arg min cCg n i1 n ķ1 1l ! Y pjq i ¤Y pjq k ) ¡ r F n,¡i,j pY pjq k |X i q & 2 , j 1, 2,
where C g is a grid of values of c and r F n,¡i,j py|xq :

°n k1,k$i K c px ¡ X k q1l tY pjq k ¤yu °n k1,k$i K c px ¡ X k q . We select the sequence h from the condition n h p k | log c| q nc p Ñ 0 by taking h c ¢ k n 1{p | log c| ¡ξ
, where ξp ¡ q and c : minpc 1 , c 2 q.

For each distribution, we simulate N 500 samples of size n 1000, and we consider several positions ty t : pt{10, 1 ¡ t{10q; t 1, . . . , 9u. Since all stable tail dependence functions satisfy maxpt, 1 ¡ tq ¤ Lpt, 1 ¡ tq ¤ 1, all the estimators have been corrected so that they satisfy these bounds. However, the estimators have not been forced to be convex although this could have been done for instance by using a constrained spline smoothing method [START_REF] Hall | Distribution and dependence-function estimation for bivariate extreme-value distributions[END_REF], or a projection technique (Fils-Villetard et al., 2008). In all the settings, C g t0.06, 0.12, 0. L k py|xq as a function of k. Each of these figures shows the behaviour of our estimator at the positions y ty t , t 1, . . . , 9u for a given value of the covariate (x 0.2, 0.5 and 0.8, respectively). Based on these simulations, we can conclude that in general, our estimator behaves well for not too large values of k with a good proximity to the true value, while some bias appears for k large, which can be expected from our theoretical results, since for k large α x pn{kq is not necessary negligible. The estimates obtained for the asymmetric logistic model show more bias than those for the bivariate Student distribution, since α x ptq converges faster to zero as t Ñ V for the latter. Indeed, for the bivariate Student distribution with ν 1 we have α x ptq t ¡2 while α x ptq t ¡1 for the asymmetric logistic distribution. From the figures it also seems that the estimation is more difficult for y close to the diagonal.

In Figures 7 and8 we show the summary performance measures Abias and MSE for Model 1 and Model 2, respectively, as a function of k for each of the covariate positions. As is clear from these figures, the performance measures do not critically depend on the position in the covariate space for k not too large. The results also seem to indicate that the estimator performs better for stronger dependence than for weaker dependence.

Application to air pollution data

In this section, the proposed methodology is applied to a dataset of air pollution measurements.

Being able to analyse the dependence between temperature and ozone concentration is of pri- mary importance in order to identify population health effects of high ozone concentration and extreme temperature. The dataset contains daily measurements on, among others, maximum temperature and ground level ozone concentration, for the time period 1999 to 2013, collected at stations spread over the U.S. by the United States Environmental Protection Agency (EPA). It is publicly available at https:{{aqsdr1.epa.gov{aqsweb{aqstmp{airdata{download files.html. We estimate the stable tail dependence function conditional on time and location, where the latter is expressed by latitude and longitude. The estimation method is the same as the one described in Section 3.1 apart from the dimension of the covariate space, namely here p 3 which implies ξ 1.1{3. We use the same grid values C g for the cross-validation, after standardising the covariates to the interval r0, 1s. As kernel function K ¦ we use the following generalisation of the bi-quadratic kernel K :

K ¦ px 1 , x 2 , x 3 q : 3 ¹ i1 Kpx i q,
where x 1 , x 2 , x 3 , refer to the covariates time, latitude and longitude, respectively, in standardised form. Note that K ¦ has as support the unit ball with respect to the max-norm on R 3 .

We report here only the results at two different time points: January 15, 2007 and June 15, 2007 in California. California has one of the largest economies in the world and as such there is a high emission of air pollutants. First, Figure 9 represents the stations in California as markers with different colors corresponding to the value of the estimates mediant s L k p0.5, 0.5|xq, k n{4, . . . , n{2u of Lp0.5, 0.5|xq. The range over which the median is computed can be motivated from the simulations, see e.g. Figures 1 till 6. Clearly, the extremal dependence between daily maximum temperature and ground level ozone concentration varies a lot across measurement stations. This could be explained by the fact that the climate of California varies widely, from hot desert to subarctic, depending on the location. As is also clear from Figure 9, the extremal dependence also varies over time. In order to get a better idea of the extremal dependence between temperature and ozone, we show in Figure 10 the time plot of the estimates of the conditional extremal coefficient ηpxq : 2 ¢ Lp0.5, 0.5|xq r1, 2s at two specific stations, Fresno and Los Angeles. This coefficient is often used in the literature as a summary measure of the extremal dependence, with perfect dependence corresponding to the value 1 and independence to the value 2. These two cities exhibit a different extremal dependence throughout the year.

Indeed, for Fresno the extremal dependence is strong in the winter months but becomes weaker in summer, while the opposite holds for Los Angeles. To get a more detailed picture of the extremal dependence we show in Figure 11 In order to prove Lemma 2.1, we only need to verify that

E p
T k py|x 0 q % Ñ f px 0 qLpy|x 0 q and Var ¡ p

T k py|x 0 q © Ñ 0 as n Ñ V.

We have

E p T k py|x 0 q % » S K Kpuq n k P A n{k,y |X x 0 ¡ hu ¨f px 0 ¡ huqdu » S K KpuqLpy|x 0 ¡ huqf px 0 ¡ huqdu » S K Kpuq ¡ n k P A n{k,y |X x 0 ¡ hu ¨¡ Lpy|x 0 ¡ huq © f px 0 ¡ huqdu. (6)
Since u S K , for n large enough, using the continuity of f and L at x 0 IntpS X q non-empty, we have boundedness in a neighborhood of x 0 and thus and by the first order condition sup

sup uS K Lpy|x 0 ¡ huq V and sup uS K f px 0 ¡ huq V,
uS K § § § n k P A n{k,y |X x 0 ¡ hu ¨¡ Lpy|x 0 ¡ huq § § § Ñ 0,
as n Ñ V.

Note that

» K 1, and hence by Lebesgue's dominated convergence theorem, we obtain, for n Ñ V,

» S K KpuqLpy|x 0 ¡ huqf px 0 ¡ huqdu Ñ f px 0 qLpy|x 0 q, and » S K Kpuq ¡ n k P A n{k,y |X x 0 ¡ hu ¨¡ Lpy|x 0 ¡ huq © f px 0 ¡ huqdu Ñ 0,
implying the first statement.

Next,

Var ¡ p T k py|x 0 q © 1 k 4 h ¡p » S K K 2 puq n k P A n{k,y |X x 0 ¡ hu ¨f px 0 ¡ huqdu B ¡ 1 n pfpx 0 qLpy|x 0 q op1qq 2 1 kh p }K} 2
2 f px 0 qLpy|x 0 q op1q ¨¡ 1 n pfpx 0 qLpy|x 0 q op1qq 2 , which goes to zero since kh p Ñ V.

Proof of Lemma 2.2

Clearly, we have

kh p Cov ¡ p T k py|x 0 q, p T k py I |x 0 q © » S K K 2 puq n k P A n{k,y A n{k,y I |X x 0 ¡ hu ¨f px 0 ¡ huqdu ¡ h p k n f px 0 q 2 Lpy|x 0 qLpy I |x 0 q op1q » S K K 2 puq n k P A n{k,y A n{k,y I |X x 0 ¡ hu ¨f px 0 ¡ huqdu op1q.
Then, we easily deduce that

P A n{k,y A n{k,y I |X x 0 ¡ hu ¨ P A n{k,y |X x 0 ¡ hu ¨ P A n{k,y I |X x 0 ¡ hu ¡ P A n{k,y A n{k,y I |X x 0 ¡ hu ¨.
Naturally we can describe the sets A n{k,y and A n{k,y I as a finite union like for any y R d

A n{k,y d ¤ j1 4 1 ¡ F j pY pjq |Xq ¤ k n y j B : d ¤ j1
A n{k,y j ,j .

Thus, we have

A n{k,y A n{k,y I d ¤ j1 3 A n{k,y j ,j A n{k,y I j ,j A d ¤ j1 4 1 ¡ F j pY pjq |Xq ¤ k n py j y I j q B d ¤ j1
A n{k,y j y I j ,j A n{k,yy I, which implies that

P A n{k,y A n{k,y I |X x 0 ¡ hu ¨ P A n{k,y |X x 0 ¡ hu ¨ P A n{k,y I |X x 0 ¡ hu ¡ P A n{k,yy I |X x 0 ¡ hu ¨,
and using the same arguments as in the proof of Lemma 2.1, the result follows.

Proof of Theorem 2.1

As a first step we consider the process

3 c kh p ¡ p T k py|x 0 q ¡ f px 0 q Lpy|x 0 q α x 0 ¡ n k © M x 0 pyq %© , y r0, T s d A ,
and study its weak convergence. Based on the decomposition

c kh p ¡ p T k py|x 0 q ¡ f px 0 q Lpy|x 0 q α x 0 ¡ n k © M x 0 pyq %© c kh p ¡ p T k py|x 0 q ¡ E p T k py|x 0 q %© c kh p ¡ E p T k py|x 0 q % ¡ f px 0 q Lpy|x 0 q α x 0 ¡ n k © M x 0 pyq %© ,
we have that the main task is to study the weak convergence of the process

3 c kh p ¡ p T k py|x 0 q ¡ E p T k py|x 0 q %© , y r0, T s d A , (7)
since

lim nÑV sup yr0,T s d c kh p § § §E p T k py|x 0 q % ¡ f px 0 q Lpy|x 0 q α x 0 ¡ n k © M x 0 pyq % § § § 0.
Indeed, if we look at [START_REF] De Carvalho | Spectral density ratio models for multivariate extremes[END_REF] in the proof of Lemma 2.1

E p T k py|x 0 q % f px 0 qLpy|x 0 q O ph η f η L q » S K Kpuq ¡ n k P A n{k,y |X x 0 ¡ hu ¨¡ Lpy|x 0 ¡ huq © f px 0 ¡ huqdu,
where the big O term is independent from y. Then

n k P A n{k,y |X x 0 ¡ hu ¨¡ Lpy|x 0 ¡ huq α x 0 ¡hu ¡ n k © 6 8 7 M x 0 ¡hu pyq ! n k P A n{k,y |X x 0 ¡ hu ¨¡ Lpy|x 0 ¡ huq α x 0 ¡hu pn{kq ¡ M x 0 ¡hu pyq ( ) D F E , where sup yr0,T s d § § § § § § n k P A n{k,y |X x 0 ¡ hu ¨¡ Lpy|x 0 ¡ huq α x 0 ¡hu pn{kq ¡ M x 0 ¡hu pyq § § § § § § ¤ sup yr0,T s d ,xBx 0 prq § § § § § § n k P A n{k,y |X x ¨¡ Lpy|xq α x pn{kq ¡ M x pyq § § § § § § Ñ 0,
combining the second order condition with the fact that for n large enough x 0 ¡ hu B x 0 prq.

This leads to

n k P A n{k,y |X x 0 ¡ hu ¨¡ Lpy|x 0 ¡ huq α x 0 ¡hu ¡ n k © pM x 0 ¡hu pyq op1qq α x 0 ¡ n k © pM x 0 ¡hu pyq op1qq ¡ α x 0 ¡hu ¡ n k © ¡ α x 0 ¡ n k ©© pM x 0 ¡hu pyq op1qq ,
where the little o component doesn't depend on y. Now,

• according to the Hölder condition on α and }u} ¤ 1 sup t¥0 |α x 0 ¡hu ptq ¡ α x 0 ptq| ¤ M α }hu} ηα O ph ηα q ,

• by uniform continuity of px, yq Ñ M x pyq over B x 0 prq ¢ r0, T s d sup yr0,T s d |M x 0 ¡hu pyq ¡ M x 0 pyq| Ñ 0 as n Ñ V.

Hence, we can deduce that n k

P A n{k,y |X x 0 ¡ hu ¨¡ Lpy|x 0 ¡ huq α x 0 ¡ n k © M x 0 pyq α x 0 ¡ n k © op1q O ph ηα q , which implies that c kh p § § §E p T k py|x 0 q % ¡ f px 0 q Lpy|x 0 q α x 0 ¡ n k © M x 0 pyq % § § § O ¡ c kh p h minpη f ,η L ,ηαq © c kh p α x 0 ¡ n k © op1q Ñ 0.
Define now the covering number N pF, L 2 pQq, τ q as the minimal number of L 2 pQq-balls of radius τ needed to cover the class of functions F and the uniform entropy integral as

Jpδ, F, L 2 q : » δ 0 log sup QQ N pF, L 2 pQq, τ }F} Q,2 q dτ,
where Q is the set of all probability measures Q for which 0 }F} 2 Q,2 :

³ F 2 dQ V and F is an envelope function for the class F.

Let P be the distribution measure of pY, Xq, and denote the expected value under P , the empirical version and empirical process as follows

P f : » f dP, P n f : 1 n n i1 f pY i , X i q , G n f : c npP n ¡ P qf,
for any real-valued measurable function f :

R d ¢ R p Ñ R.
We introduce our sequence of classes F n on R d ¢ R p as

F n :
3 pu, zq Ñ f n,y pu, zq, y r0, T s d A where f n,y pu, zq :

n k h p K h px 0 ¡ zq1l t1¡F 1 pu 1 |zq¤k{n y 1 or ... or 1¡F d pu d |zq¤k{n y d u .
Denote also by F n an envelope function of the class F n . Now, according to Theorem 19.28 in van der Vaart (1998), the weak convergence of the stochastic process [START_REF] Coles | Dependence measures for extreme value analyses[END_REF] follows from the following four conditions. Let ρ x 0 be a semimetric, possibly depending on x 0 , making r0, T s d totally bounded. We have to prove that sup ρx 0 py,y I q¤δn P pf n,y ¡ f n,y Iq 2 ÝÑ 0 for every δ n × 0,

P F 2 n Op1q, (8)
P F 2 n 1l tFn¡ε c nu ÝÑ 0 for every ε ¡ 0, (9)
Jpδ n , F n , L 2 q ÝÑ 0 for every δ n × 0.

(

) 11
We start with proving (8). We have

P pf n,y ¡ f n,y Iq 2 » S K Kpuq 2 n k E ¡ 1l A n{k,y ¡ 1l A n{k,y I © 2 |X x 0 ¡ hu & f px 0 ¡ huqdu. But, for any x I S X E ¡ 1l A n{k,y ¡ 1l A n{k,y I © 2 § § X x I & P A n{k,y A n{k,y I § § X x I ¨¡ P A n{k,y A n{k,y I § § X x I
P tA n{k,y A n{k,y IuztA n{k,y A n{k,y Iu § § X x I ¨. [START_REF] Dutang | Robust and bias-corrected estimation of the coefficient of tail dependence[END_REF] Using the same notation as in the proof of Lemma 2.2, we have

A n{k,y A n{k,y I 5 d ¤ j1 A n{k,y j ,j C £ 5 d ¤ j1 A n{k,y I j ,j C d ¤ j1 d ¤ i1 3 A n{k,y j ,j A n{k,y I i ,i A d ¤ j1 3 A n{k,y j ,j A n{k,y I j ,j A d ¤ j1
A n{k,y j y I j ,j .

29

Then, with A c denoting the complement of any set A, it follows that tA n{k,y A n{k,y IuztA n{k,y A n{k,y Iu Returning now to [START_REF] Dutang | Robust and bias-corrected estimation of the coefficient of tail dependence[END_REF], we have

P tA n{k,y A n{k,y IuztA n{k,y A n{k,y Iu § § X x I ¤ d j1 P ¢ k n py j y I j q ¤ 1 ¡ F j pY pjq |Xq ¤ k n py j y I j q § § X x I ¤ k n d j1 |y j ¡ y I j |.
Thus, defining

ρ x 0 py, y I q d j1 |y j ¡ y I j |,
also called the Manhattan distance on R d , which is clearly a semimetric making r0, T s d totally bounded, we have proven [START_REF] Daouia | On kernel smoothing for extremal quantile regression[END_REF].

We define now the envelope functions

F
P F 2 n » S K Kpuq 2 n k P A n{k,y T |X x 0 ¡ hu ¨f px 0 ¡ huqdu Lpy T |x 0 qfpx 0 q}K} 2 2 op1q.
For [START_REF] Dekkers | A moment estimator for the index of an extreme-value distribution[END_REF], note that we have tF n ¡ ε c nu tpF n {ε c nq ξ ¡ 1u for any ξ ¡ 0, thus

P F 2 n 1l tFn¡ε c nu ¤ 1 ε ξ n ξ{2 P F 2 ξ n 1 ε ξ n ξ{2 ¡ n kh p © ξ{2 » S K Kpuq 2 ξ n k PpA n{k,y T |X x 0 ¡ huqf px 0 ¡ huqdu ¡ ε c kh p
where the right-hand side converges towards 0 kh p Ñ V and K satisfies Assumption pK 1 q.

Finally, it remains to prove [START_REF] Drees | Best attainable rates of convergence for estimators of the stable tail dependence function[END_REF]. Define the following class of functions on R d ¢ R p r F :

N p r F n , L 2 pQq, τ } r F n } Q,2 q ¤ CV p16eq V ¢ 1 τ 2pV ¡1q
.

Next, we retrieve F n by multiplying the previous family with one single function, i.e.

F n tz Ñ nh p {kK h px 0 ¡ zqu ¢ r F n .
Since only one ball is needed to cover the class tz Ñ nh p {kK h px 0 ¡ zqu whatever the measure Q Q, according to the last inequality in the proof of Theorem 2.10.20 in van der Vaart and Wellner (1996

) sup QQ N pF n , L 2 pQq, τ }F n } Q,2 q ¤ CV p16eq V ¢ 1 τ 2pV ¡1q : L ¢ 1 τ V .
Thus, [START_REF] Drees | Best attainable rates of convergence for estimators of the stable tail dependence function[END_REF] is established since for any δ n × 0 and n large enough, we have

Jpδ n , F n , L 2 q ¤ » δn 0 logpLq ¡ V logpτ qdτ op1q.
Finally, we consider the process (3). Straightforward calculations give the following decomposition

c kh p £ p T k py|x 0 q p f n px 0 q ¡ Lpy|x 0 q ¡ α x 0 ¡ n k © M x 0 pyq c kh p f px 0 q ¡ p T k py|x 0 q ¡ f px 0 q Lpy|x 0 q α x 0 ¡ n k © M x 0 pyq %© ¡ c kh p p T k py|x 0 q p f n px 0 qfpx 0 q p p f n px 0 q ¡ f px 0 qq.
Note that for the first term in the right-hand side of the above display we have just established the weak convergence as a stochastic process (apart from the factor 1{f px 0 q), whereas for the second term we have essentially to study c kh p p p f n px 0 q ¡ f px 0 qq. The latter can be rewritten as

c kh p p p f n px 0 q ¡ f px 0 qq k n c nh p p p f n px 0 q ¡ f px 0 qq.
Under our assumptions on K and f one can easily verify that c nh p p p f n px 0 q ¡ f px 0 qq O P p1q

(see e.g. [START_REF] Parzen | On estimation of a probability density function and mode[END_REF]) and hence the theorem follows.

Hence, sup θΘ c nP G n pθ, a n q ¤ 2dn h p k r n a n pfpx 0 q op1qq Ñ 0, [START_REF] Escobar-Bach | Bias-corrected and robust estimation of the bivariate stable tail dependence function[END_REF] and the assertion follows.

Proof We have that r n n{k Ñ 0 since n h p {k r n converges and kh p Ñ V, and as such ( 16) is established.

By the first order condition, we have

P E 2 n » S K Kpuq 2 n k P A n{k,y T pn{kqrnya |X x 0 ¡ hu ¨f px 0 ¡ huqdu Lpy T |x 0 qfpx 0 q}K} 2 2 op1q,
where y a : pa, . . . , aq R d and ( 17) follows. Now we verify condition [START_REF] Gannoun | Reference ranges based on nonparametric quantile regression[END_REF]. For any ξ ¡ 0 we obtain the following inequality

P E 2 n 1l tEn¥ c nu ¤ 1 ε ξ n ξ{2 P E 2 ξ n ¤ 1 ε ξ d pkh p q ξ{2 ¡ T r n n k a
© p}K} 2 ξ 2 ξ f px 0 q op1qq, which tends to zero under the assumptions of the theorem.

It remains to show that

Jpd n , tG n pθ, aq : θ Θu , L 2 q ÝÑ 0 for all d n × 0.

To deal with the uniform entropy integral, we can reuse the lines of proof of T k py|x 0 q ¡ p T k py|x 0 q % § § § op1q.

According to the notation in the proof of Theorem 2.2, note that T k py|x 0 q p f n px 0 q ¡ Lpy|x 0 q ¡ α x 0 ¡ n k © M x 0 pyq , y r0, T s d

c kh p E § § § q T k ¡ p T k § § § % py|x 0 q equals c n E § § § § § 1 n n i1 n k h p K h px 0 ¡ X i q1l ! 1¡ p F n,
T k ¡ p T k § § § % py|x 0 q ¤ sup

C

, can be handled using the same arguments as those at the end of the proof of Theorem 2.1.

and

  Huang (1998), de Haan and Ferreira (2006) and Bücher et al. (2014). We also refer to Peng (2010), Fougères et al. (2015), Beirlant et al. (2016) and Escobar-Bach et al. (2017b) where

(

  2014b) developed a nonparametric and asymptotically unbiased estimator based on weighted exceedances over a high threshold. The extension of this regression estimation of tail parameters to the full max-domain of attraction, has been considered in Daouia et al. (2013), who generalized Daouia et al. (2011), and also by Stupfler (2013) and Goegebeur et al. (2014a)

3. 1 A

 1 small simulation study Our aim in this section is to illustrate the finite sample behaviour of our estimator s L k py|xq : q T k py|xq p f n pxq with a small simulation study. We focus on dimension d 2 and we consider the two following models: Model 1: We consider the bivariate Student distribution with density function

  [START_REF] Capéraà | Bivariate distributions with given extreme value attractor[END_REF] and also mentioned in[START_REF] Fougères | Bias correction in multivariate extremes[END_REF], namely:

  h and c have to be chosen. Concerning c, we can use the following cross validation criterion introduced by Yao (1999), implemented by Gannoun et al. (2002), and already used in an extreme value context by Daouia et al. (2011, 2013) or Goegebeur et al. (2015):

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :

 1234567 Figure 1: Model 1: Sample mean of s L k py|0.2q as a function of k with, from left to right and up to down, y y t , t 1, . . . , 9, respectively. The true value of the parameter is represented by a full horizontal black line.

Figure 8 :

 8 Figure 8: Model 2: Absolute bias and MSE as a function of k for different covariate positions x 0.2 (full line), 0.5 (dashed line), 0.8 (dotted line).

Figure 9 :

 9 Figure 9: Air pollution data: Estimates mediant s L k p0.5, 0.5|xq, k n{4, . . . , n{2u of Lp0.5, 0.5|xq for stations in California on January 15, 2007 (left) and June 15, 2007 (right).

Figure 10 :

 10 Figure 10: Air pollution data: Time plot of the estimate 2 ¢ mediant s L k p0.5, 0.5|xq, k n{4, . . . , n{2u of the conditional extremal coefficient at Fresno (left) and Los Angeles (right) over the year 2007.

Figure 11 :

 11 Figure 11: Air pollution data: Estimate of the Pickands dependence function mediant s L k pt, 1 ¡ t|xq, k n{4, . . . , n{2u for Fresno (top) and Los Angeles (bottom) on January 15, 2007 (first column) and June 15, 2007 (second column).

3 A 4 kn

 34 n{k,y j y I j ,j A c n{k,y j y I py j y I j q ¤ 1 ¡ F j pY pjq |Xq ¤ k n py j y I j q B .

2 u

 2 Theorem 2.1 by considering the following class of functions on r0, 1s d Ñ 1l ty 1 ¤1¡u 1 ¤y 2 or ... or y 2d¡1 ¤1¡u d ¤y 2d u , y 1 y 2 , . . . , y 2d¡1 y 2d @ , which is a V C-class since the class of sets associated to the indicator functions is a V C-class of sets as the union of d V C-classes of sets. This allows us to prove that there exist positive constants C and V such that sup QQ N ptG n pθ, aq : θ Θu , L 2 pQq, τ }E n } Q,2 q ¤ C last assertion follows. This achieves the proof of Theorem 2.2.

yr0,T s dc

  nP G n py, aq Ñ 0, by Assertion 1 since it is clear that a n Ñ 0 can be replaced by any fixed value a in[START_REF] Escobar-Bach | Bias-corrected and robust estimation of the bivariate stable tail dependence function[END_REF] and conclude with the fact that nr n h p {k Ñ 0.

  j 1, . . . , d, we denote by F j p.|xq, the continuous conditional distribution function of Y pjq given X x and Lp.|xq the conditional stable tail dependence function defined as lim F 1 pY p1q |Xq ¤ t ¡1 y 1 or . . . or 1 ¡ F d pY pdq |Xq ¤ t ¡1 y d |X x

tÑV tP ¡ 1 ¡ © Lpy 1 , . . . , y d |xq. (1)

  Lemma 2.2 Under the assumptions of Lemma 2.1, we have for any y, y I R d

	kh p Cov	¡	p

  18, 0.24, 0.3u and ξ 1.1 are used as chosen in Escobar-Bach et al. (2017a).An extensive simulation study has also indicated that these choices seem to give always reason-able results. Concerning the kernel, each time we use the bi-quadratic function Kpxq : 15 16 p1 ¡ x 2 q 2 1l r¡1,1s pxq. As a qualitative measure of the efficiency over the different positions ty t , t 1, . . . , 9u, we define the absolute bias and the mean squared error (MSE) respectively as follows

	Abiaspx, kq :	1 9N	9 ţ1	N i1	§ § § s L piq k py t |xq ¡ Lpy t |xq	§ § §
	MSEpx, kq :	1 9N	9 ţ1	N i1	¡ L s piq k py t |xq ¡ Lpy t |xq	© 2

. Figures 1-3 (respectively Figures 4-6) represent the sample means in case of Model 1 (respectively Model 2), based on N samples of size n, of our estimator s

  n pu, zq : K h px 0 ¡ zq1l t1¡F 1 pu 1 |zq¤k{n T or ... or 1¡F d pu d |zq¤k{n T u .

	n k h p With y T : pT, . . . , T q loooomoooon , assertion (9) results from Lemma 2.1 since
	d times

  3 pu, zq Ñ 1l t1¡F 1 pu 1 |zq¤y 1 or ... or 1¡F d pu d |zq¤y d u , y R d A 3 pu, zq Ñ 1l t1¡ ¤ ¤y 1 or ... or 1¡ ¤ ¤y d u ¥ pF 1 , . . . , F d qpu, zq, y R d A .Let's focus for a moment on the class of functions on r0,1s d 3 u Ñ 1l t1¡u 1 ¤y 1 or ... or 1¡u d ¤y d u , y R d A .Since this is a family of indicator functions, it is a V C-class if and only if the family of sets associated to the indicator functions is a V C-class of sets. The latter sets can be easily represented as the union of d V C-classes of sets and thus it is also a V C-class of sets (see Lemma 2.6.17 (iii) pu, zq Ñ 1l t1¡F 1 pu 1 |zq¤k{n y 1 or ... or 1¡F d pu d |zq¤k{n y d u , y r0, T s d A

in van der

[START_REF] Van Der Vaart | Weak convergence and empirical processes, with applications to statistics[END_REF]

. Next, according to Lemma 2.

6.18 (vii)

in van der Vaart and Wellner (1996), it follows that r F is a V C-class with V C-index V fixed. Define now r F n : 3 , and the envelope function r F n pu, zq : 1l t1¡F 1 pu 1 |zq¤k{n T or ... or 1¡F d pu d |zq¤k{n T u . The previous arguments for r F remain, thus we have that r F n is also a V C-class with V C-index V . According to Theorem 2.6.7 in van der Vaart and Wellner (1996), there exists a universal constant C such that for Q the set of all probability measures on R d ¢ R p and any 0 τ 1 sup QQ

  K h px 0 ¡ zq1l t1¡δ 1 pu 1 ,zq¤k{n θ 1 or ... or 1¡δ d pu d ,zq¤k{n θ d u . |δ n ¡δ 0 | converges in probability towards the null function H 0 : t0u in H endowed with the norm }δ} H : °d i1 }δ i } V . In order to apply Theorem 2.3 in van der Vaart and Wellner (2007), pθ, a n q ÝÑ 0 for every a n Ñ 0, Assertion 2: sup θΘ |G n G n pθ, aq| P 0, for every a ¡ 0, where G n pθ, aq is an envelope function for the class G n pθ, aq : tg θ,δ 0 rnδ,n ¡ g θ,δ 0 ,n : δ H, }δ} H ¤ au . Proof of Assertion 1. Using the ideas of the proof of Theorem 2.1, for any δ H such that }δ} H ¤ a |q θ,δ 0 rnδ,n ¡ q θ,δ 0 ,n |pu, zq ¤ 1l tk{n θ 1 ¡rna¤1¡F 1 pu 1 |zq¤k{n θ 1 rna or ... or k{n θ d ¡rna¤1¡F d pu d |zq¤k{n θ d rnau : 1l B n,θ,a pu, zq. Thus, we set G n pθ, aqpu, zq : n k h p K h px 0 ¡ zq1l B n,θ,a pu, zq and we have c nP G n pθ, a n q n KpuqP pB n,θ,an |X x 0 ¡ hu q f px 0 ¡ huqdu, with P pB n,θ,an |X x 0 ¡ hu q ¤ ¡ r n a n ¤ 1 ¡ F j pY pjq |Xq ¤ k{n θ j r n a n |X x 0 ¡ hu

	4.4 Proof of Theorem 2.2 For convenience, denote δ n : ¡ p F n,1 , . . . , p F n,d we have now to show Assertion 1: sup θΘ c » nP G n h p k S K

Let

I n : tg θ,δ,n : θ Θ, δ Hu, where for θ Θ : r0, T s d , and δ H : tδ pδ 1 , . . . , δ d q; δ j : R ¢ S X Ñ Ru with g θ,δ,n pu, zq :

n k h p K h px 0 ¡ zqq θ,

δ,n pu, zq : n k h p © and δ 0 : pF 1 , . . . , F d q. According to Lemma 2.3, r ¡1 n d j1 P ¡ k{n θ j © ¤ 2dr n a n .

  of Assertion 2. The idea is to apply Lemma 2.2 in van der Vaart and Wellner (2007).Now we work with the class of functions2 G n pθ, aq, θ r0, T s d @, for any a ¡ 0 with the envelope functionE n pu, zq : n k h p K h px 0 ¡ zq1l t1¡F 1 pu 1 |zq¤k{n T rna or ... or 1¡F d pu d |zq¤k{n T rnau .For what concerns condition (16), we have according[START_REF] Escobar-Bach | Local robust estimation of the Pickands dependence function[END_REF] P G n pθ, aq 2 P pB n,θ,a |X x 0 ¡ hu q f px 0 ¡ huqdu

	» ¤ 2d n S K k 2d n k	Kpuq 2 n k r n a » S K r n a f px 0 q}K} 2 Kpuq 2 f px 0 ¡ huqdu 2 op1q ¨.
	Consequently, we have first to prove that
		sup θΘ	P G n pθ, aq 2 ÝÑ 0,	(16)
			P E 2 n Op1q,	(17)
	P E 2 n 1l tEn¥ c nu Ñ 0 for every ε ¡ 0.	(18)

  1 pY p1q i |X i q¤k{n y 1 or ... or 1¡ p F n,d pY pdq K h px 0 ¡ X i q1l ! 1¡F 1 pY p1q i |X i q¤k{n y 1 or ... or 1¡F d pY pdq E r|g y,δn,n pY 1 , X 1 q ¡ g y,δ 0 ,n pY 1 , X 1 q|s ¤ c n P G n py, aq,for n large enough, since with probability tending to 1, δ n δ 0 r n Bp0, aq where Bp0, aq : tδ : }δ} H ¤ au, and by using the Skorohod representation. This implies that

				i	|X i q¤k{n y d	)
	¡ ¤ c n sup n k h p i yr0,T s d c kh p E § § § q	|X i q¤k{n y d	)	& § & § § §

Acknowledgement

This work was supported by a research grant (VKR023480) from VILLUM FONDEN and an international project for scientific cooperation (PICS-6416). The authors sincerely thank the editor, associate editor and the referees for their helpful comments and suggestions that led to substantial improvement of the paper.