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HAL Id: hal-03132477

https://hal.archives-ouvertes.fr/hal-03132477

Preprint submitted on 5 Feb 2021

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Robust estimation of the conditional stable tail

dependence function

Yuri Goegebeur, Armelle Guillou, Jing Qin

To cite this version:

Yuri Goegebeur, Armelle Guillou, Jing Qin. Robust estimation of the conditional stable tail depen-dence function. 2021. �hal-03132477�

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Robust estimation of the conditional stable tail dependence

function

Yuri Goegebeurp1q, Armelle Guilloup2q and Jing Qinp1q

p1q Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej

55, 5230 Odense M, Denmark

p2q Institut Recherche Math´ematique Avanc´ee, UMR 7501, Universit´e de Strasbourg et CNRS, 7 rue

Ren´e Descartes, 67084 Strasbourg cedex, France

Abstract

We propose a robust estimator of the stable tail dependence function in the case where random covariates are recorded. Under suitable assumptions, we derive the finite dimensional weak convergence of the estimator properly normalized.

Keywords: Multivariate extreme value statistics; Stable tail dependence function; Local estimation; Robustness; Empirical processes.

MSC 2010 subject classifications: Primary 62G32, 62G35; Secondary 62G05, 62G20.

1

Introduction

A topic of central interest in multivariate extreme values is to measure the strength of depen-dence in the extremes. This can be done by using some coefficients of tail dependepen-dence or some functions, among them the Pickands dependence function or the stable tail dependence func-tion. In the present paper, we focus on this latter function introduced by Huang (1992), and we estimate it when the random variables of main interest are recorded along with random covariates, related to the target variables. That means that we are in the regression context where our objective is to estimate the stable tail dependence function between the response variables conditional on the covariates. This leads to the concept of conditional stable tail de-pendence function. Additionally, since in practice some outliers may occur in real datasets with a disturbing effect on the estimates of dependencies, we propose estimators which are robust against observations that are atypical for the extreme dependence structure of the models under consideration. In other words, our goal in this paper is to propose a robust estimator of the conditional stable tail dependence function. This topic has been partially considered in the recent literature, e.g., by Escobar-Bach et al. (2017) where a robust estimator for the stable tail dependence function was introduced, but not in a regression context or by Escobar-Bach et al. (2018b) who considered estimation of the conditional stable tail dependence function, though not in a robust way. Other works on estimation of extremal dependence with covariates include, e.g., Gardes and Girard (2015), Mhalla et al. (2019), Escobar-Bach et al. (2020). Note that in Goegebeur et al. (2021), a nonparametric robust estimator of the conditional tail dependence

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coefficient has been proposed, but this topic is much simpler than the one considered in the present paper, since now we have to estimate a function rather than a single parameter. Concretely, throughout the paper, we denote by pYp1q, Yp2qq a bivariate random vector recorded

along with a random covariate X P Rd. Let }.} denote some norm on Rd and Bxprq the closed

ball with respect to }.} centered at x and radius r ą 0. For j “ 1, 2, we denote by Fjp.|xq, the

continuous conditional distribution function of Ypjq given X “ x, by f

X the density function

of the covariate X and by x0 a reference position such that x0 P IntpSXq, the interior of the

support SX Ă Rd, which is assumed to be non-empty. Our aim in this paper is to estimate the

conditional stable tail dependence function defined as lim rÑ8rP ´ 1 ´ F1pYp1q|Xq ď r´1y1 or 1 ´ F2pYp2q|Xq ď r´1y2 ˇ ˇ ˇX “ x ¯ “ Lpy1, y2|xq

in a robust way, where we assume that the above limit exists for all x P SX.

To reach this goal, we assume that, for all x P SX, our bivariate random vector pYp1q, Yp2qq

satisfies the model P ´ 1 ´ F1pYp1q|Xq ď y1, 1 ´ F2pYp2q|Xq ď y2 ˇ ˇ ˇX “ x ¯ “ yd11pxqy d2pxq 2 gpy1, y2|xq p1 ` δpy1, y2|xqq , (1)

for any py1, y2q P r0, 1s2ztp0, 0qu, where d1pxq, d2pxq are positive and continuous functions such

that d1pxq ` d2pxq “ 1, gpy1, y2|xq is continuous in py1, y2, xq and homogeneous of order 0 in

py1, y2q, and δpy1, y2|xq is a function of constant sign in the neighbourhood of zero, with |δp., .|xq|

a bivariate regularly varying function, that is, there exists a function ξp., .|xq such that lim

rÓ0

|δ|pry1, ry2|xq

|δ|pr, r|xq “ ξpy1, y2|xq,

for all py1, y2q P r0, 8q2ztp0, 0qu, where the convergence is uniform in py1, y2q P p0, T s2 and

x P Bx0pζq, for any T ą 0 and ζ ą 0. Also, ξpy1, y2|xq is assumed to be continuous in py1, y2, xq

and homogeneous of order βpxq ą 0 in py1, y2q. The approach followed will be non-parametric

and based on local estimation of extreme value models in a neighborhood of the point of interest in the covariate space.

As for obtaining robustness, the density power divergence criterion initially proposed by Basu et al. (1998) will be used. It is defined between two density functions f and h as follows

∆αpf, hq :“ $ ’ ’ ’ ’ & ’ ’ ’ ’ % ż R „ h1`αpyq ´ ˆ 1 ` 1 α ˙ hαpyqf pyq ` 1 αf 1`α pyq  dy, α ą 0, ż R logf pyq hpyqf pyqdy, α “ 0.

Here, f is assumed to be the true (typically unknown) density of the data, whereas h is a parametric model, depending on a vector of parameters which is estimated by minimizing the

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empirical version of ∆αpf, hq. This estimator is called the minimum density power divergence

(MDPD) estimator. Note also that, when α is equal to zero or one, one recovers the Kullback-Leibler or L2´ divergences, respectively. This tuning parameter α needs to be selected in order to ensure a trade-off between efficiency and robustness of the MDPD estimator. A value of α close to 0 leads to an efficient but not robust estimator (α “ 0 corresponding to the maximum likelihood estimator), whereas a larger value of α increases the robustness of the estimator while decreasing its efficiency.

Our paper is organized as follows. In Section 2, we assume that both conditional marginal dis-tribution functions are known, and we propose a robust estimator of the conditional stable tail dependence function for which we establish the finite dimensional weak convergence. Then, in Section 3, we consider the more realistic situation where the conditional marginal distribution functions are unknown. We estimate again in a robust way the conditional stable tail depen-dence function and we derive similar results as in the previous section, under some additional assumptions. All the proofs are postponed to the Appendix.

2

Estimation of Lpy

1

, y

2

|x

0

q in case of known margins

For convenience, assume that the conditional marginal distributions F1p.|xq and F2p.|xq are unit

Pareto and let Zt:“ mintYp1q,1´tt Yp2qu for t :“ y1y`y1 2, 0 ă t ă 1.

Then, according to model (1), the conditional survival function of Zt given X “ x, denoted by

FZtp.|xq, is a conditional Pareto-type model of the following form:

FZtpy|xq “ Ktpxqy ´1 p1 ` δtpy|xqq , (2) where Ktpxq :“ ˆ t 1 ´ t ˙d2pxq g ˆ 1, t 1 ´ t ˇ ˇ ˇx ˙ , δtpy|xq :“ δ ˆ 1 y, t 1 ´ t 1 y ˇ ˇ ˇx ˙ .

Note that δtp.|xq is regularly varying at infinity with index ´βpxq, i.e., δtpuy|xq{ δtpu|xq Ñ y´βpxq

as u Ñ 8 for all y ą 0. Additionally, we assume in the sequel the following classical condition. Assumption pDtq For all x P SX, the conditional survival function of Zt given by (2) is such

that |δtp.|xq| is normalized regularly varying with index ´βpxq, i.e.,

δtpy|xq “ Ctpxq exp ˆży 1 εtpu|xq u du ˙ ,

with Ctpxq P R and εtpy|xq Ñ ´βpxq as y Ñ 8. Moreover, we assume y Ñ εtpy|xq to be a

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We now turn to the estimation of Lpy1, y2|x0q. The above implies that Lpy1, y2|x0q “ lim ∆Ó0 1 ∆ ! P ´ 1 ´ F1pYp1q|x0q ď ∆ y1 ˇ ˇ ˇX “ x0 ¯ ` P ´ 1 ´ F2pYp2q|x0q ď ∆ y2 ˇ ˇ ˇX “ x0 ¯ ´P ´ 1 ´ F1pYp1q|x0q ď ∆ y1, 1 ´ F2pYp2q|x0q ď ∆ y2 ˇ ˇ ˇX “ x0 ¯) “ y1` y2´ lim ∆Ó0 1 ∆P ˆ Z1´tě 1 ∆ y1 ˇ ˇ ˇX “ x0 ˙ “ y1` y2´ y1K1´tpx0q.

Thus, estimating the conditional stable tail dependence function requires the estimation of K1´tpx0q. To reach this goal, note that from (2), we deduce that

K1´tpx0q “ k n UZ1´tpn{k|x0q 1 ` δ1´tpUZ1´tpn{k|x0q|x0q ,

where UZ1´tp.|x0q is the conditional tail quantile function defined as UZ1´tp.|x0q :“ infty :

FZ1´tpy|x0q ě 1 ´ 1{.u, and k is an intermediate sequence such that k Ñ 8 and k{n Ñ 0. If

p

UZ1´tp.|x0q and pδ1´tp.|x0q are estimators for UZ1´tp.|x0q and δ1´tp.|x0q, respectively, then by the

plug-in method we derive the following estimator for K1´tpx0q:

p K1´t,kpx0q :“ k n p UZ1´tpn{k|x0q 1 ` pδ1´tp pUZ1´tpn{k|x0q|x0q , (3)

which yields a simple estimator for the conditional stable tail dependence function: p

Lkpy1, y2|x0q :“ y1` y2´ y1Kp1´t,kpx0q. (4) Recall that we want to propose a robust estimator. To this aim, we will adjust the MDPD criterion to the local estimation context. Remark that FZ1´t belongs to the class of distribution

functions of Beirlant et al. (2009). Thus, the distribution of the relative excesses Z1´t{un

given Z1´tą un can, for un large, be approximated by an extended Pareto distribution (EPD)

function given by Hpy; δ1´tpun|x0q, βpx0qq :“ # 1 ´ y´1 ” 1 ` δ1´tpun|x0q ´ 1 ´ y´βpx0q ¯ı´1 , y ą 1, 0, y ď 1,

where δ1´tpun|x0q ą maxt´1, ´1{βpx0qu. Moreover, using Proposition 2.3 in Beirlant et al.

(2009) the approximation error is uniformly opδ1´tpun|x0qq for un Ñ 8. Using this property,

one can estimate δ1´tpun|x0q with the MDPD approach as follows.

Starting from pYip1q, Yip2q, Xiq, i “ 1, . . . , n, independent copies of pYp1q, Yp2q, Xq, we obtain

pZ1´t,i, Xiq, i “ 1, . . . , n, independent copies of pZ1´t, Xq, and fit the density function h

associ-ated with H and defined for y ą 1 as h py; δ1´tpun|x0q, βpx0qq :“ y´2 ” 1 ` δ1´tpun|x0q ´ 1 ´ y´βpx0q ¯ı´2 ˆ ” 1 ` δ1´tpun|x0q ´ 1 ´ p1 ´ βpx0qqy´βpx0q ¯ı ,

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locally to the relative excesses Z1´t,i{ pUZ1´tpn{k|x0q, i “ 1, . . . , n, given that Z1´t,ią pUZ1´tpn{k|x0q.

Here, pUZ1´tpn{k|x0q is the natural estimator for UZ1´tpn{k|x0q defined as pUZ1´tpn{k|x0q :“

infty : pFZ1´tpy|x0q ě 1 ´ k{nu where, for pFZ1´tpy|x0q, we use the kernel-type estimator

p

FZ1´tpy|x0q :“

1 n

řn

i“1Khnpx0´ Xiq1ltZ1´t,iďyu

1 n řn i“1Khnpx0´ Xiq , (5) with Khnp.q :“ Kp.{hnq{h d

n, K a joint density on Rd and hn a positive non-random sequence

satisfying hnÑ 0 as n Ñ 8.

This leads to the minimum density power divergence estimator, pδn,1´t:“ pδ1´tp pUZ1´tpn{k|x0q|x0q,

for δ1´tp pUZ1´tpn{k|x0q|x0q, and defined as the point minimizing the empirical density power

divergence, that is, for α ą 0 p ∆α,1´tpδ1´t|x0q :“ 1 k n ÿ i“1 Khnpx0´ Xiq # ż8 1 h1`αpy; δ1´t, βqdy ´ ˆ 1 ` 1 α ˙ hα ˜ Z1´t,i p UZ1´tpn{k|x0q ; δ1´t, β ¸+ 1ltZ 1´t,ią pUZ1´tpn{k|x0qu.

In our proposed procedure, we only estimate δ1´tp pUZ1´tpn{k|x0q|x0qq with the density power

divergence criterion, while β will be fixed, either at the true value or a mis-specified one. Fixing this second order rate parameter β at some value is quite common when fitting second order models like (2) to data, see, e.g., Feuerverger and Hall (1999), Gomes and Martins (2004), Du-tang et al. (2014), Escobar-Bach et al. (2017).

The minimization is based on the derivative of the empirical density power divergence. Direct computations show that all the terms appearing in this derivative have the following form

Sn,1´tps|x0q :“ 1 k n ÿ i“1 Khnpx0´ Xiq ˜ Z1´t,i p UZ1´tpn{k|x0q ¸s 1ltZ 1´t,ią pUZ1´tpn{k|x0qu for s ă 0.

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Assuming FZ1´tpy|x0q is strictly increasing in y, we can rewrite this main statistic as follows: Sn,1´tps|x0q “ 1 k n ÿ i“1 Khnpx0´ Xiq # 1 ` żZ1´t,i p UZ1´tpn{k|x0q s us´1 p UZs1´tpn{k|x0q du + 1l tZ1´t,ią pUZ1´tpn{k|x0qu “ 1 k n ÿ i“1 Khnpx0´ Xiq1ltZ1´t,ią pU Z1´tpn{k|x0qu `1 k n ÿ i“1 Khnpx0´ Xiq # żZ1´t,i p UZ1´tpn{k|x0q s us´1 p Us Z1´tpn{k|x0q du + 1ltZ 1´t,ią pUZ1´tpn{k|x0qu “ 1 k n ÿ i“1 Khnpx0´ Xiq1ltZ1´t,ią pU Z1´tpn{k|x0qu ` ż8 p UZ1´tpn{k|x0q 1 k n ÿ i“1 Khnpx0´ Xiq s us´1 p UZs 1´tpn{k|x0q 1ltuăZ1´t,iudu “ 1 k n ÿ i“1 Khnpx0´ Xiq1ltZ1´t,ią pU Z1´tpn{k|x0qu ` ż8 p UZ1´tpn{k|x0q 1 k n ÿ i“1 Khnpx0´ Xiq s us´1 p Us Z1´tpn{k|x0q 1ltF Z1´tpZ1´t,i|x0qănknkFZ1´tpu|x0qudu “ 1 k n ÿ i“1 Khnpx0´ Xiq1ltZ1´t,ią pU Z1´tpn{k|x0qu ` ż1 0 1 k n ÿ i“1 Khnpx0´ Xiqsz ´1´s1l tFZ1´tpZ1´t,i|x0qăknnkFZ1´tpz´1UpZ1´tpn{k|x0q|x0qudz “ Tn,1´tpsn,1´tp1|x0q|x0q ` ż1 0 Tn,1´tpsn,1´tpz|x0q|x0q s z´1´sdz, (6) where Tn,1´tpy|x0q :“ 1 k n ÿ i“1 Khnpx0´ Xiq1ltFZ1´tpZ1´t,i|x0qăknyu, y P p0, T s, sn,1´tpz|x0q :“ n kFZ1´t ´ z´1UpZ1´tpn{k|x0q ˇ ˇ ˇx0 ¯ .

To establish the weak convergence of Sn,1´tps|x0q, we need to assume some classical conditions

due to the regression context, which are nowadays well-known in the conditional extreme value framework.

First, the density fX and the functions appearing in FZ1´tpy|xq need to satisfy the following

H¨older conditions.

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ηC1´t and ηε1´t, such that for all x, z P SX: |fXpxq ´ fXpzq| ď MfX}x ´ z} ηfX, |K1´tpxq ´ K1´tpzq| ď MK1´t}x ´ z} ηK1´t , |C1´tpxq ´ C1´tpzq| ď MC1´t}x ´ z} ηC1´t , sup yě1 |ε1´tpy|xq ´ ε1´tpy|zq| ď Mε1´t}x ´ z} ηε1´t.

Then, we have also to impose a condition on the kernel function K, which is a standard condition in local estimation.

Assumption pK1q K is a bounded density function on Rd, with support SK included in the unit

ball in Rd.

We have now all the ingredients to state the joint weak convergence of Sn,1´tjpsj|x0q, j “

1, . . . , M . Note that we allow for the possibility that tj “ tj1 for j ‰ j1, but of course the

statistics Sn,1´tjpsj|x0q, j “ 1, . . . , M , must be different. This is due to the fact that, for a given

value of t, the study of the MDPD estimator pδn,1´trequires the joint convergence in distribution

of several statistics Sn,1´tps|x0q, with different values of s. In the sequel, weak convergence is

denoted by .

Theorem 2.1 Assume pD1´tjq and pH1´tjq for j “ 1, . . . , M , pD0.5q, pH0.5q, pK1q, x0 P

IntpSXq with fXpx0q ą 0, and y Ñ FZ1´tjpy|x0q, j “ 1, . . . , M , are strictly increasing. Consider

sequences k Ñ 8 and hnÑ 0 as n Ñ 8 such that k{n Ñ 0, khdnÑ 8, h

ηε1´t1^¨¨¨^ηε1´t M^ηε0.5 n lognk Ñ 0, akhd nh ηfXK1´t1^¨¨¨^ηK1´t M n Ñ 0, a khd n|δ1´tjpUZ1´tjp n k|x0q|x0q| Ñ 0, j “ 1, . . . , M . Then,

for n Ñ 8, we have, for s1, . . . , sM ă 0,

b khd n ¨ ˚ ˝ Sn,1´t1ps1|x0q ´ 1 1´s1fXpx0q .. . Sn,1´tMpsM|x0q ´ 1 1´sMfXpx0q ˛ ‹ ‚ ¨ ˚ ˚ ˚ ˝ s1 ş1 0 ”W 1´t1pzq z ´ W1´t1p1q ı z´s1dz .. . sM ş1 0 ”W 1´tMpzq z ´ W1´tMp1q ı z´sMdz ˛ ‹ ‹ ‹ ‚ ,

where W1´tjpyq, j “ 1, . . . , M, are zero centered Gaussian processes with

EpW1´tjpyqW1´tj1pyqq “ }K}

2 2fXpx0q ˜ max ˜ K1´tjpx0q y , K1´tj1px0q y ¸¸´d1px0q ˆ ˜ max ˜ tj 1 ´ tj K1´tjpx0q y , tj1 1 ´ tj1 K1´tj1px0q y ¸¸´d2px0q ˆ g ¨ ˚ ˚ ˝ 1 max ˆ K1´tjpx0q y , K1´t j1px0q y ˙ , 1 max ˆ tj 1´tj K1´tjpx0q y , tj1 1´tj1 K1´t j1px0q y ˙ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ x0 ˛ ‹ ‹ ‚ .

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Note that in case tj “ tj1, the covariance in Theorem 2.1 simplifies to

E`W1´tjpyqW1´tjpyq

˘

“ }K}22fXpx0q py ^ yq .

The next theorem gives the joint asymptotic behavior of the MDPD estimator pδn,1´tj, j “

1, . . . , J .

Theorem 2.2 Under the conditions of Theorem 2.1, with probability tending to one, there exists sequences of solutions ppδn,1´tjqně1, j “ 1, . . . , J, to the MDPD estimating equations such that

¨ ˚ ˝ p δn,1´t1 ´ δ1´t1p pUZ1´t1pn{k|x0q|x0q .. . p δn,1´tJ ´ δ1´tJp pUZ1´tJpn{k|x0q|x0q ˛ ‹ ‚ P Ñ 0.

Moreover, for the consistent solution sequences one has that

b khd n ¨ ˚ ˝ p δn,1´t1´ δ1´t1p pUZ1´t1pn{k|x0q|x0q .. . p δn,1´tJ´ δ1´tJp pUZ1´tJpn{k|x0q|x0q ˛ ‹ ‚ c ¨ ˚ ˚ ˚ ˝ 2αş10”W1´t1pzq z ´ W1´t1p1q ı z2αdz ´ p1 ` βpx 0qqp2α ` βpx0qq ş1 0 ”W 1´t1pzq z ´ W1´t1p1q ı z2α`βpx0qdz .. . 2αş10”W1´tJpzq z ´ W1´tJp1q ı z2αdz ´ p1 ` βpx 0qqp2α ` βpx0qq ş1 0 ”W 1´tJpzq z ´ W1´tJp1q ı z2α`βpx0qdz ˛ ‹ ‹ ‹ ‚ , where c :“ p1 ` 2αqp1 ` 2α ` βpx0qqp1 ` 2α ` 2βpx0qq β2px 0qp1 ` βpx0q ` 4α2` 2αβpx0qq 1 fXpx0q .

The next theorem states the main result of this section namely the joint weak convergence of the estimators pLkpy1,j, y2,j|x0q, j “ 1, . . . , J , after proper normalization.

Theorem 2.3 Under the conditions of Theorem 2.1, we have that b khd n ¨ ˚ ˝ p Lkpy1,1, y2,1|x0q ´ Lpy1,1, y2,1|x0q .. . p Lkpy1,J, y2,J|x0q ´ Lpy1,J, y2,J|x0q ˛ ‹ ‚ ¨ ˚ ˝ L1 .. . LJ ˛ ‹ ‚, where Lj :“ ´y1,jK1´tjpx0q W1´tjp1q fXpx0q ` y1,jK1´tjpx0qc " 2α ż1 0 „ W1´tjpzq z ´ W1´tjp1q  z2αdz ´p1 ` βpx0qqp2α ` βpx0qq ż1 0 „ W1´tjpzq z ´ W1´tjp1q  z2α`βpx0qdz * , j “ 1, . . . , J.

In practice, the conditional marginal distribution functions F1p.|x0q and F2p.|x0q are unknown.

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3

Estimation of Lpy

1

, y

2

|x

0

q in case of unknown margins

We consider the general framework where F1p.|xq and F2p.|xq are unknown conditional margins.

We want to mimic what has been done in the previous section by transforming the margins into approximate unit Pareto distributions. To this aim, we define

q Z1´t:“ min # 1 1 ´ pFn,1pYp1q|Xq ,1 ´ t t 1 1 ´ pFn,2pYp2q|Xq + ,

where the estimators pFn,j, j “ 1, 2, are defined as

p Fn,jpy|x0q :“ 1 n řn i“1Kcnpx0´ Xiq1ltYpjq i ďyu 1 n řn i“1Kcnpx0´ Xiq ,

with cn a positive non-random sequence satisfying cnÑ 0 as n Ñ 8. Note that this estimator

has the same form as (5) but with a bandwidth cn which needs to be different from hn (see

Theorem 3.1 below) and the kernel used here is, for simplicity, the same as the one used in the MDPD method.

Then, we introduce the same key statistic as Sn,1´tps|x0q but, this time, defined with Z1´t

replaced by qZ1´t, that is q Sn,1´tps|x0q :“ 1 k n ÿ i“1 Khnpx0´ Xiq ˜ q Z1´t,i p UZq 1´tpn{k|x0q ¸s 1lt qZ 1´t,ią pUZ1´tq pn{k|x0qu .

Similarly as in (6), assuming that FZ1´tpy|x0q is strictly increasing in y, we can show that

q Sn,1´tps|x0q “ qTn,1´tpqsn,1´tp1|x0q|x0q ` ż1 0 q Tn,1´tpqsn,1´tpz|x0q|x0q s z´1´sdz, where q Tn,1´tpy|x0q :“ 1 k n ÿ i“1 Khnpx0´ Xiq1ltFZ1´tp qZ1´t,i|x0qăknyu, y P p0, T s, q sn,1´tpz|x0q :“ n kFZ1´t ´ z´1Up q Z1´tpn{k|x0q ˇ ˇ ˇx0 ¯ .

To establish the weak convergence of qSn,1´tps|x0q, we need to impose again some assumptions,

in particular a H¨older-type condition on each marginal conditional distribution function Fj,

j “ 1, 2.

Assumption pFmq. There exist MFj ą 0 and ηFj ą 0 such that |Fjpy|xq ´ Fjpy|zq| ď

MFj}x ´ z}

ηFj

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Concerning the kernel K a stronger assumption than pK1q is needed.

Assumption pK2q. K satisfies Assumption pK1q, there exists δ, m ą 0 such that B0pδq Ă SK

and Kpuq ě m for all u P B0pδq, and K belongs to the linear span (the set of finite

lin-ear combinations) of functions k ě 0 satisfying the following property: the subgraph of k, tps, uq : kpsq ě uu, can be represented as a finite number of Boolean operations among sets of the form tps, uq : qps, uq ě ϕpuqu, where q is a polynomial on Rdˆ R and ϕ is an arbitrary real function.

The latter assumption has already been used in Gin´e and Guillou (2002) or Gin´e et al. (2004). As stated in these contributions, it is satisfied by Kpxq “ φtapxqu, a being a polynomial and φ a bounded real function of bounded variation (see, e.g., Nolan and Pollard, 1987). This is also the case, e.g., if the graph of K is a pyramid (truncated or not), or if K “ 1lr´1,1sp, etc.

Then we can show the following joint weak convergence of qSn,1´tjpsj|x0q, j “ 1, . . . , M .

Theorem 3.1 Assume that there exists b ą 0 such that fXpxq ě b, @x P SX Ă Rd, fX

is bounded, pD1´tjq, pH1´tjq for j “ 1, . . . , M , pD0.5q, pH0.5q, pK2q, pFmq hold, and that

y Ñ FZ1´tjpy|x0q, j “ 1, . . . , M, are strictly increasing at x0 P IntpSXq non-empty.

Con-sider sequences k Ñ 8, hn Ñ 0 and cn Ñ 0 as n Ñ 8, such that k{n Ñ 0, khdn Ñ 8,

hηε1´t1^¨¨¨^ηε1´tM^ηε0.5 n lognk Ñ 0, a khd nh ηfXK1´t1^¨¨¨^ηK1´t M n Ñ 0, a khd n|δ1´tjpUZ1´tjp n k|x0q|x0q| Ñ

0, j “ 1, . . . , M . Assume also that there exists an ε ą 0 such that for n sufficiently large inf

xPSX

λ ptu P B0p1q : x ´ cnu P SXuq ą ε, (7)

where λ denotes the Lebesgue measure, and for some q ą 1 and 0 ă η ă minpηF1, ηF2q

n d hdn k max ˜d | log cn|q ncd n , cηn ¸ ÝÑ 0, as n Ñ 8. (8)

Then, we have for s1, . . . , sM ă 0,

b khd n ¨ ˚ ˝ q Sn,1´t1ps1|x0q ´ 1 1´s1fXpx0q .. . q Sn,1´tMpsM|x0q ´ 1 1´sMfXpx0q ˛ ‹ ‚ ¨ ˚ ˚ ˚ ˝ s1 ş1 0 ”W 1´t1pzq z ´ W1´t1p1q ı z´s1dz .. . sM ş1 0 ”W 1´tMpzq z ´ W1´tMp1q ı z´sMdz ˛ ‹ ‹ ‹ ‚ ,

where the processes W1´tj, j “ 1, . . . , M , are as in Theorem 2.1.

Note that this limiting process is the same as the one obtained in Theorem 2.1. Compared to the latter theorem, the additional conditions required in Theorem 3.1 are needed to measure the discrepancy between the conditional distribution function Fjpy|xq and its empirical kernel

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We come back now to our final goal, that is, the robust estimation of our conditional stable tail dependence function in case of unknown margins. A similar estimator as the one defined in (4) can be proposed q Lkpy1, y2|x0q :“ y1` y2´ y1Kq1´t,kpx0q, where q K1´t,kpx0q :“ k n p U q Z1´tpn{k|x0q 1 ` qδ1´tp pUZq1´tpn{k|x0q|x0q ,

with qδ1´tp pUZq1´tpn{k|x0q|x0q the MDPD estimator based on qZ1´t.

Now, we have all the ingredients to establish the joint weak convergence of the estimators q

Lkpy1,j, y2,j|x0q, j “ 1, . . . , J , after proper normalization.

Theorem 3.2 Under the conditions of Theorem 3.1, we have that

b khd n ¨ ˚ ˝ q Lkpy1,1, y2,1|x0q ´ Lpy1,1, y2,1|x0q .. . q Lkpy1,J, y2,J|x0q ´ Lpy1,J, y2,J|x0q ˛ ‹ ‚ ¨ ˚ ˝ L1 .. . LJ ˛ ‹ ‚,

where Lj, j “ 1, . . . , J, are defined as in Theorem 2.3.

4

Appendix

4.1 Auxiliary results in case of known margins

First we establish the joint weak convergence of processes Wn,1´tj :“ t

a khd

nrTn,1´tjpy|x0q ´

yfXpx0qs; y P p0, T su, j “ 1, . . . , J .

Lemma 4.1 Assume pD1´tjq and pH1´tjq for j “ 1, . . . , J , pD0.5q, pH0.5q, pK1q, x0 P IntpSXq

with fXpx0q ą 0, and y Ñ FZ1´tjpy|x0q, j “ 1, . . . , J , are strictly increasing. Consider sequences

k Ñ 8 and hn Ñ 0 as n Ñ 8 such that k{n Ñ 0, khdn Ñ 8, h

ηε1´t1^¨¨¨^ηε1´t J^ηε0.5 n lognk Ñ 0, akhd nh ηfXK1´t1^¨¨¨^ηK1´t J n Ñ 0, a khd n|δ1´tjpUZ1´tjp n k|x0q|x0q|h ηC1´t j n Ñ 0, j “ 1, . . . , J , a khd n|δ1´tjpUZ1´tjp n k|x0q|x0q|h ηε1´t j

n lognk Ñ 0, j “ 1, . . . , J . Then, for n Ñ 8, we have

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in `Jpp0, T sq, for any T ą 0, where W1´t1, . . . , W1´tJ are zero centered Gaussian processes with

EpW1´tjpyqW1´tj1pyqq “ }K}

2 2fXpx0q ˜ max ˜ K1´tjpx0q y , K1´tj1px0q y ¸¸´d1px0q ˆ ˜ max ˜ tj 1 ´ tj K1´tjpx0q y , tj1 1 ´ tj1 K1´tj1px0q y ¸¸´d2px0q ˆ g ¨ ˚ ˚ ˝ 1 max ˆ K1´tjpx0q y , K1´t j1px0q y ˙ , 1 max ˆ tj 1´tj K1´tjpx0q y , tj1 1´tj1 K1´t j1px0q y ˙ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ x0 ˛ ‹ ‹ ‚ .

Lemma 4.2 Under the assumptions of Lemma 4.1, for any sequence upjqn satisfying

b khd n ¨ ˝ FZ1´tjpUZ1´tjpn{k|x0q|x0q FZ1´tjpupjqn |x0q ´ 1 ˛ ‚Ñ cj P R, as n Ñ 8, j “ 1, . . . , J , we have ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˝ b nhd nFZ1´t1pup1qn |x0q ˜ p FZ1´t1pup1qn |x0q FZ1´t1pup1qn |x0q´ 1 ¸ .. . b nhd nFZ1´tJpupJ qn |x0q ˜ p FZ1´t Jpu pJ q n |x0q FZ1´t Jpu pJ q n |x0q´ 1 ¸ ˛ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‚ 1 fXpx0q ¨ ˚ ˝ W1´t1p1q .. . W1´tJp1q ˛ ‹ ‚.

Lemma 4.3 Assume pD1´tjq and pH1´tjq for j “ 1, . . . , J , pD0.5q, pH0.5q, pK1q, x0 P IntpSXq

with fXpx0q ą 0, and y Ñ FZ1´tjpy|x0q, j “ 1, . . . , J , are strictly increasing. Consider sequences

k Ñ 8 and hnÑ 0 as n Ñ 8 such that k{n Ñ 0, khdn Ñ 8, h

ηε1´t1^¨¨¨^ηε1´t J^ηε0.5 n lognk Ñ 0, a khd nh ηfXK1´t1^¨¨¨^ηK1´t J n Ñ 0, a khd n|δ1´tjpUZ1´tjp n k|x0q|x0q| Ñ 0, j “ 1, . . . , J . Then, we have b khd n ¨ ˚ ˚ ˚ ˚ ˝ p UZ1´t1pn{k|x0q UZ1´t1pn{k|x0q´ 1 .. . p UZ1´t Jpn{k|x0q UZ1´t Jpn{k|x0q ´ 1 ˛ ‹ ‹ ‹ ‹ ‚ 1 fXpx0q ¨ ˚ ˝ W1´t1p1q .. . W1´tJp1q ˛ ‹ ‚.

4.2 Proof of the auxiliary results in case of known margins

Proof of Lemma 4.1. From Corollary 5.1 in Goegebeur et al. (2020), with a slight adjust-ment for the left continuity of Tn,1´tjpy|x0q, we have that Wn,1´tj W1´tj in `pp0, T sq, for

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j “ 1, . . . , J . To obtain the joint convergence of pWn,1´t1, . . . , Wn,1´tJq we need to verify the

finite dimensional convergence and joint tightness. We start by proving the finite dimensional convergence. Let Tn:“ » — – Tn,1´t1py1|x0q .. . Tn,1´tJpyJ|x0q fi ffi fl and T :“ » — – y1fXpx0q .. . yJfXpx0q fi ffi fl. Since b khd npTn´ T q “ b khd npTn´ EpTnqq ` b khd npEpTnq ´ T q, andakhd

npEpTnq ´ T q Ñ 0, by Proposition 5.1 in Goegebeur et al. (2020) and our assumptions,

it is sufficient to study the weak convergence of akhd

npTn´ EpTnqq. To this aim we use the

Cram´er-Wold device, and show that Λn:“ ψT

b khd

npTn´ EpTnqq N p0, ψTΣψq

for all ψ :“ pψ1, . . . , ψJqT P RJ, where the elements of Σ are as in the statement of Lemma 4.1.

By a straightforward rearrangement of terms we have

Λn “ n ÿ i“1 c hd n k J ÿ j“1 ψj „ Khnpx0´ Xiq1ltFZ1´t jpZ1´tj ,i|x0qă k nyju´ E ˆ Khnpx0´ Xq1ltFZ1´t jpZ1´tj|x0qă k nyju ˙ “: n ÿ i“1 Vi,n.

Since V1,n, . . . , Vn,n are independent and identically distributed random variables, we have

VarpΛnq “ nVarpV1,nq, and hence

VarpΛnq “ J ÿ j“1 J ÿ j1“1 ψjψj1 nhdn k Cj,j1, (9) where Cj,j1 :“ Cov ˆ Khnpx0´ Xq1ltFZ1´t jpZ1´tj|x0qă k nyju , Khnpx0´ Xq1ltFZ1´t j1pZ1´tj1|x0qă k nyj1u ˙ . We have E ˆ Kh2npx0´ Xq1ltF Z1´tjpZ1´tj|x0qă k nyj,FZ1´tj1pZ1´tj1|x0qă k nyj1u ˙ “ E ˆ Kh2npx0´ XqP ˆ FZ1´tjpZ1´tj|x0q ă k nyj, FZ1´tj1pZ1´tj1|x0q ă k nyj1 ˇ ˇ ˇ ˇ X ˙˙ . Now let mp1qn :“ max ˆ UZ1´tj ˆ n kyj ˇ ˇ ˇ ˇx0 ˙ , UZ1´t j1 ˆ n kyj1 ˇ ˇ ˇ ˇx0 ˙˙ , mp2qn :“ max ˆ tj 1 ´ tj UZ1´tj ˆ n kyj ˇ ˇ ˇ ˇx0 ˙ , tj1 1 ´ tj1 UZ1´t j1 ˆ n kyj1 ˇ ˇ ˇ ˇx0 ˙˙ .

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Then, for x P SX, P ˆ FZ1´tjpZ1´tj|x0q ă k nyj, FZ1´tj1pZ1´tj1|x0q ă k nyj1 ˇ ˇ ˇ ˇ X “ x ˙ “ P ´ Yp1q ą mp1qn , Yp2q ą mp2qn ˇ ˇ ˇ X “ x ¯ . We have

UZ1´tpy|x0q “ K1´tpx0qyr1 ` a1´tpy|x0qs,

where a1´tp.|x0q is regularly varying with index ´βpx0q, and hence

mp1q n “ n kmax ˜ K1´tjpx0q yj „ 1 ` a1´tj ˆ n kyj ˇ ˇ ˇ ˇ x0 ˙ ,K1´tj1px0q yj1 „ 1 ` a1´tj1 ˆ n kyj1 ˇ ˇ ˇ ˇ x0 ˙¸ “: n kmr p1q n , mp2qn “ n kmax ˜ tj 1 ´ tj K1´tjpx0q yj „ 1 ` a1´tj ˆ n kyj ˇ ˇ ˇ ˇx0 ˙ , tj1 1 ´ tj1 K1´tj1px0q yj1 „ 1 ` a1´tj1 ˆ n kyj1 ˇ ˇ ˇ ˇx0 ˙¸ “: n kmr p2q n .

Using (1) gives then P ´ Yp1q ą mp1qn , Yp2q ą mp2qn ˇ ˇ ˇ X “ x ¯ “ k npmr p1q n q´d1pxqpmr p2q n q´d2pxqg ˜ 1 r mp1qn , 1 r mp2qn ˇ ˇ ˇ ˇ ˇ x ¸ « 1 ` δ ˜ k n 1 r mp1qn ,k n 1 r mp2qn ˇ ˇ ˇ ˇ ˇ x ¸ff , and thus E ˆ Kh2npx0´ Xq1ltF Z1´tjpZ1´tj|x0qă k nyj,FZ1´t j1pZ1´tj1|x0qă k nyj1u ˙ “ k nhd n ż SK K2pvqpmr p1q n q´d1px0´hnvqpmr p2q n q´d2px0´hnvqg ˜ 1 r mp1qn , 1 r mp2qn ˇ ˇ ˇ ˇ ˇ x0´ hnv ¸ ˆ « 1 ` δ ˜ k n 1 r mp1qn ,k n 1 r mp2qn ˇ ˇ ˇ ˇ ˇ x0´ hnv ¸ff fXpx0´ hnvqdv. Write δ ˜ k n 1 r mp1qn ,k n 1 r mp2qn ˇ ˇ ˇ ˇ ˇ x0´ hnv ¸ “ δˆ k n, k n ˇ ˇ ˇ ˇ x0 ˙ δ`k n, k n ˇ ˇx0´ hnv ˘ δ`kn,nkˇˇx0 ˘ « ξ ˜ 1 r mp1qn , 1 r mp2qn ˇ ˇ ˇ ˇ ˇ x0´ hnv ¸ ` δ ´ k n 1 r mp1qn ,nk 1 r mp2qn ˇ ˇ ˇ x0´ hnv ¯ δ`kn,nkˇˇx0´ hnv ˘ ´ ξ ˜ 1 r mp1qn , 1 r mp2qn ˇ ˇ ˇ ˇ ˇ x0´ hnv ¸ fi ffi fl.

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By (1), the assumptions of Lemma 4.1, and the fact that δ0.5p.|xq satisfies pD0.5q and pH0.5q, one has that δ ˜ k n 1 r mp1qn ,k n 1 r mp2qn ˇ ˇ ˇ ˇ ˇ x0´ hnv ¸ “ δˆ k n, k n ˇ ˇ ˇ ˇ x0 ˙« ξ ˜ 1 r mp1qn , 1 r mp2qn ˇ ˇ ˇ ˇ ˇ x0´ hnv ¸ ` op1q ff ,

where the op1q term is uniform in v P SK. Hence, by Lebesgue’s dominated convergence theorem

E ˆ Kh2npx0´ Xq1ltF Z1´tjpZ1´tj|x0qănkyj,FZ1´t j1 pZ1´tj1|x0qăknyj1u ˙ “ k nhd n $ & % }K}22fXpx0q ˜ max ˜ K1´tjpx0q yj ,K1´tj1px0q yj1 ¸¸´d1px0q ˆ ˜ max ˜ tj 1 ´ tj K1´tjpx0q yj , tj1 1 ´ tj1 K1´tj1px0q yj1 ¸¸´d2px0q ˆ g ¨ ˚ ˚ ˝ 1 max ˆ K1´tjpx0q yj , K1´t j1px0q yj1 ˙ , 1 max ˆ tj 1´tj K1´tjpx0q yj , tj1 1´tj1 K1´t j1px0q yj1 ˙ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ x0 ˛ ‹ ‹ ‚` op1q , / / . / / -.

By taking into account that

E ˆ Khnpx0´ Xq1ltFZ1´t jpZ1´tj|x0qă k nyju ˙ “ k n E`Tn,1´tjpyj|x0q ˘ “ k n yjfXpx0qp1 ` op1qq,

where the last step follows from Proposition 5.1 in Goegebeur et al. (2020), we have that nhdn k Cj,j1 Ñ }K} 2 2fXpx0q ˜ max ˜ K1´tjpx0q yj ,K1´tj1px0q yj1 ¸¸´d1px0q ˆ ˜ max ˜ tj 1 ´ tj K1´tjpx0q yj , tj1 1 ´ tj1 K1´tj1px0q yj1 ¸¸´d2px0q ˆ g ¨ ˚ ˚ ˝ 1 max ˆ K1´tjpx0q yj , K1´t j1px0q yj1 ˙ , 1 max ˆ tj 1´tj K1´tjpx0q yj , tj1 1´tj1 K1´t j1px0q yj1 ˙ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ x0 ˛ ‹ ‹ ‚ ,

which shows the convergence of the variance in (9).

To ensure the convergence in distribution of Λn to a normal random variable, we have to verify

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In the present context this simplifies to showing that nE|V1,n|3 Ñ 0. We have E|V1,n|3 ď ˆ hd n k ˙3{2 $ & % E » – ˜ J ÿ j“1 |ψj|Khnpx0´ Xq1ltFZ1´t jpZ1´tj|x0qă k nyju ¸3fi fl `3E » – ˜ J ÿ j“1 |ψj|Khnpx0´ Xq1ltFZ1´t jpZ1´tj|x0qă k nyju ¸2fi fl ˆE ˜ J ÿ j“1 |ψj|Khnpx0´ Xq1ltFZ1´t jpZ1´tj|x0qă k nyju ¸ `4 « E ˜ J ÿ j“1 |ψj|Khnpx0´ Xq1ltFZ1´t jpZ1´tj|x0qă k nyju ¸ff3, . -.

With arguments similar to the ones used above when studying the covariance terms, one finds that E|V1,n|3 “ Op1{pn

a khd

nqq, and hence nE|V1,n|3Ñ 0. This establishes the finite dimensional

convergence. The joint tightness follows from the individual tightness (similarly to Lemma 1 in

Bai and Taqqu, 2013). 

Proof of Lemma 4.2. Note that p FZ1´tjpupjqn |x0q FZ1´tjpupjqn |x0q “ 1 1 n řn i“1Khnpx0´ Xiq 1 n řn i“1Khnpx0´ Xiq1ltZ 1´tj ,iąupjqn u FZ1´tjpupjqn |x0q “ 1 1 n řn i“1Khnpx0´ Xiq FZ1´tj ´ UZ1´tjpn{k|x0q|x0 ¯ FZ1´tjpupjqn |x0q Tn,1´tj ´n kFZ1´tj ´ upjq n |x0 ¯ˇ ˇ ˇx0 ¯ .

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Then, by using Cram´er-Wold device, we have for all ψ :“ pψ1, . . . , ψJqT P RJ J ÿ j“1 ψj c nhd nFZ1´tjpu pjq n |x0q » – p FZ1´tJpupjqn |x0q FZ1´tJpupjqn |x0q ´ 1 fi fl “ J ÿ j“1 ψj c nhd nFZ1´tjpupjqn |x0q ˆ » – 1 1 n řn i“1Khnpx0´ Xiq FZ1´tj ´ UZ1´tjpn{k|x0q|x0 ¯ FZ1´tjpupjqn |x0q Tn,1´tj ´n kFZ1´tj ´ upjq n |x0 ¯ˇ ˇ ˇx0 ¯ ´ 1 fi fl “ J ÿ j“1 ψj a khd n ” Tn,1´tj ´ n kFZ1´tj ´ upjqn |x0 ¯ˇ ˇ ˇx0 ¯ ´nkFZ1´tjpupjqn |x0qfXpx0q ı ´ W1´tj ´ n kFZ1´tj ´ upjqn |x0 ¯¯ 1 n řn i“1Khnpx0´ Xiq ` J ÿ j“1 ψj b khd n ¨ ˚ ˝ g f f f e FZ1´tj ´ UZ1´tjpn{k|x0q|x0 ¯ FZ1´tjpupjqn |x0q ´ 1 ˛ ‹ ‚ Tn,1´tj ´ n kFZ1´tj ´ upjqn |x0 ¯ˇ ˇ ˇx0 ¯ ´ nkFZ1´tjpupjqn |x0qfXpx0q 1 n řn i“1Khnpx0´ Xiq ` J ÿ j“1 ψj W1´tj ´ n kFZ1´tj ´ upjqn |x0 ¯¯ ´ W1´tjp1q 1 n řn i“1Khnpx0´ Xiq ` J ÿ j“1 ψj W1´tjp1q 1 n řn i“1Khnpx0´ Xiq ´ c k n a nhd n “1 n řn i“1Khnpx0´ Xiq ´ fXpx0q ‰ 1 n řn i“1Khnpx0´ Xiq J ÿ j“1 ψj g f f f e FZ1´tjpupjqn |x0q FZ1´tj ´ UZ1´tjpn{k|x0q|x0 ¯ J ÿ j“1 ψj W1´tjp1q fXpx0q ,

under our assumptions combined with Lemma 4.1 and the Skorohod construction.  Proof of Lemma 4.3. First, remark that

P ˜ b khd n ˜ p UZ1´tj pn{k|x0q UZ1´tj pn{k|x0q ´ 1 ¸ ď zj, @j “ 1, . . . , J ¸ “ P ¨ ˝ c nhd nFZ1´tjpapjqn |x0q ¨ ˝ p FZ1´tjpapjqn |x0q FZ1´tjpapjqn |x0q ´ 1 ˛ ‚ ď c nhd nFZ1´tjpapjqn |x0q ¨ ˝ FZ1´tjpUZ1´tjpnk|x0q|x0q FZ1´tjpapjqn |x0q ´ 1 ˛ ‚, @j “ 1, . . . , J ˛ ‚, where apjqn :“ UZ1´tjpn{k|x0qp1 ` zj{ a khd

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for all j “ 1, . . . , J , we have b khd n ¨ ˝ FZ1´tjpUZ1´tjpn{k|x0q|x0q FZ1´tjpapjqn |x0q ´ 1 ˛ ‚ ÝÑ cj P R.

This convergence holds since, under our assumptions, we have b khd n ¨ ˝ FZ1´tjpUZ1´tjpn{k|x0q|x0q FZ1´tjpapjqn |x0q ´ 1 ˛ ‚ “ zj 1 ` δ1´tjpUZ1´tjpn{k|x0q|x0q 1 ` δ1´tjpa pjq n |x0q ´ b khd n δ1´tjpUZ1´tjpn{k|x0q|x0q 1 ` δ1´tjpa pjq n |x0q ˜ δ1´tjpa pjq n |x0q δ1´tjpUZ1´tjpn{k|x0q|x0q ´ 1 ¸ ÝÑ zj. Now, since c nhd nFZ1´tjpapjqn |x0q ¨ ˝ FZ1´tjpUZ1´tjpn{k|x0q|x0q FZ1´tjpapjqn |x0q ´ 1 ˛ ‚ ÝÑ zj, by Lemma 4.2 and continuity we deduce that

P ˜ b khd n ˜ p UZ1´tjpn{k|x0q UZ1´tjpn{k|x0q ´ 1 ¸ ď zj, @j “ 1, . . . , J ¸ ÝÑ Pˆ W1´tjp1q fXpx0q ď zj, @j “ 1, . . . , J ˙ .

This achieves the proof of Lemma 4.3. 

4.3 Proof of Theorem 2.1.

First we consider the weak convergence of an individual statistic Sn,1´tps|x0q, properly

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b khd n ˆ Sn,1´tps|x0q ´ 1 1 ´ sfXpx0q ˙ “ ż1 0 rW1´tpzq ´ W1´tp1qs s z´1´sdz ` "b khd nrTn,1´tpsn,1´tp1|x0q|x0q ´ sn,1´tp1|x0qfXpx0qs ´ W1´tpsn,1´tp1|x0qq * ` tW1´tpsn,1´tp1|x0qq ´ W1´tp1qu ` b khd npsn,1´tp1|x0q ´ 1q fXpx0q ` ż1 0 "b khd nrTn,1´tpsn,1´tpz|x0q|x0q ´ sn,1´tpz|x0qfXpx0qs ´ W1´tpsn,1´tpz|x0qq * s z´1´sdz ` ż1 0 rW1´tpsn,1´tpz|x0qq ´ W1´tpzqs s z´1´sdz `fXpx0q b khd n ż1 0 rsn,1´tpz|x0q ´ zs s z´1´sdz “: ż1 0 rW1´tpzq ´ W1´tp1qs s z´1´sdz ` 6 ÿ i“1 Ti,k. (10)

We study the terms separately. Clearly, using Lemma 5.2 from Goegebeur et al. (2020) we have that for n large, with arbitrary large probability,

|T1,k| ď sup yPp0,2s ˇ ˇ ˇ ˇ b khd nrTn,1´tpy|x0q ´ yfXpx0qs ´ W1´tpyq ˇ ˇ ˇ ˇ , (11) and |T4,k| ď sup yPp0,2s ˇ ˇ ˇ ˇ b khd nrTn,1´tpy|x0q ´ yfXpx0qs ´ W1´tpyq ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ż1 0 s z´1´sdz ˇ ˇ ˇ ˇ, (12) and hence, by Lemma 4.1 combined with the Skorohod construction we obtain T1,k “ oPp1q and

T4,k “ oPp1q.

Using again Lemma 5.2 in Goegebeur et al. (2020) with continuity, we have

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Concerning T3,k, we can use the following decomposition: T3,k “ b khd n « FZ1´tp pUZ1´tpn{k|x0q|x0q FZ1´tpUZ1´tpn{k|x0q|x0q ´ 1 ff fXpx0q “ b khd n » – ˜ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q ¸´1 ´ 1 fi fl 1 ` δ1´tp pUZ1´tpn{k|x0q|x0q 1 ` δ1´tpUZ1´tpn{k|x0q|x0q fXpx0q ` b khd n « 1 ` δ1´tp pUZ1´tpn{k|x0q|x0q 1 ` δ1´tpUZ1´tpn{k|x0q|x0q ´ 1 ff fXpx0q “ b khd n » – ˜ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q ¸´1 ´ 1 fi fl 1 ` δ1´tp pUZ1´tpn{k|x0q|x0q 1 ` δ1´tpUZ1´tpn{k|x0q|x0q fXpx0q ` b khd n δ1´tpUZ1´tpn{k|x0q|x0q 1 ` δ1´tpUZ1´tpn{k|x0q|x0q $ & % » – δ1´tp pUZ1´tpn{k|x0q|x0q δ1´tpUZ1´tpn{k|x0q|x0q ´ ˜ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q ¸´βpx0qfi fl ` » – ˜ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q ¸´βpx0q ´ 1 fi fl , . -. “: ´ b khd n « p UZ1´tpn{k|x0q UZ1´tpn{k|x0q ´ 1 ff fXpx0qp1 ` oPp1qq ` b khd n δ1´tpUZ1´tpn{k|x0q|x0q 1 ` δ1´tpUZ1´tpn{k|x0q|x0q T3,kp1q. By Proposition B.1.10 in de Haan and Ferreira (2006), for n large, with arbitrary large proba-bility, we have for ε, ξ ą 0

|T3,kp1q| ď ε ˜ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q ¸´βpx0q˘ξ ` ˜ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q ¸´βpx0q ` 1. (14)

In the above, the notation a˘‚ means aif a ě 1 and a´‚ if a ă 1. This implies by Lemma 4.3

and our conditions that

T3,k ´W1´tp1q. (15)

Concerning now T5,k, we have for any δ P p0, 1q small

|T5,k| ď żδ 0 |W1´tpsn,1´tpz|x0qq ´ W1´tpzq| |s| z´1´sdz ` ż1 δ |W1´tpsn,1´tpz|x0qq ´ W1´tpzq| |s| z´1´sdz ď |s| # sup zPp0,δs |W1´tpsn,1´tpz|x0qq| ` sup zPp0,δs |W1´tpzq| + żδ 0 z´1´sdz `|s| sup zPpδ,1s |W1´tpsn,1´tpz|x0qq ´ W1´tpzq| ż1 δ z´1´sdz “ oPp1q. (16)

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Finally, concerning T6,k, we have T6,k “ fXpx0q b khd n ż1 0 « FZ1´tpz ´1 p UZ1´tpn{k|x0q|x0q FZ1´tpUZ1´tpn{k|x0q|x0q ´ z ff s z´1´sdz “ fXpx0q b khd n $ & % ˜ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q ¸´1 ´ 1 , . -s ż1 0 z´sdz `fXpx0q b khd n ˜ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q ¸´1 ż1 0 ˜ 1 ` δ1´tpz´1UpZ1´tpn{k|x0q|x0q 1 ` δ1´tpUZ1´tpn{k|x0q|x0q ´ 1 ¸ s z´sdz “: ´fXpx0q s 1 ´ s b khd n ˜ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q ´ 1 ¸ p1 ` oPp1qq `fXpx0q b khd n ˜ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q ¸´1 δ1´tpUZ1´tpn{k|x0q|x0q 1 ` δ1´tpUZ1´tpn{k|x0q|x0q T6,kp1q with |T6,kp1q| ď |s| ż1 0 ˇ ˇ ˇ ˇ ˇ ˇ δ1´tpz´1UpZ 1´tpn{k|x0q|x0q δ1´tpUZ1´tpn{k|x0q|x0q ´ ˜ z´1UpZ1´tpn{k|x0q UZ1´tpn{k|x0q ¸´βpx0qˇˇ ˇ ˇ ˇ ˇ z´sdz `|s| ż1 0 ˇ ˇ ˇ ˇ ˇ ˇ ˜ z´1UpZ1´tpn{k|x0q UZ1´tpn{k|x0q ¸´βpx0q ´ 1 ˇ ˇ ˇ ˇ ˇ ˇ z´sdz “ OPp1q,

using arguments similar to those for T3,kp1q. Consequently, using again Lemma 4.3, we deduce that

T6,k ´

s

1 ´ sW1´tp1q. (17)

Combining decomposition (10) with (11)-(17), the proof of the marginal weak convergence of Sn,1´tps|x0q, properly normalized, is achieved.

The joint weak convergence of pakhd

nrSn,1´tjpsj|x0q ´ fXpx0q{p1 ´ sjqs, j “ 1, . . . , M q follows

from Lemmas 4.1 and 4.3, respectively. 

4.4 MDPD calculations and proofs of Theorem 2.2 and Theorem 2.3

Derivatives

We need to compute the two first derivatives of the empirical divergence. Direct computations yield

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d p∆α,1´tpδ1´t|x0q dδ1´t “ p1 ` αq # ż8 1 hαpy; δ1´t, βq d dδ1´t hpy; δ1´t, βqdy 1 k n ÿ i“1 Khnpx0´ Xiq1ltZ1´t,ią pU Z1´tpn{k|x0qu ´1 k n ÿ i“1 Khnpx0´ Xiqh α´1 ˜ Z1´t,i p UZ1´tpn{k|x0q ; δ1´t, β ¸ d dδ1´t h ˜ Z1´t,i p UZ1´tpn{k|x0q ; δ1´t, β ¸ 1ltZ 1´t,ią pUZ1´tpn{k|x0qu + “ p1 ` αq # 2αβpx0q p1 ` 2αqp1 ` 2α ` βpx0qq 1 k n ÿ i“1 Khnpx0´ Xiq1ltZ1´t,ią pU Z1´tpn{k|x0qu `1 k n ÿ i“1 Khnpx0´ Xiq ˜ Z1´t,i p UZ1´tpn{k|x0q ¸´2α 1ltZ 1´t,ią pUZ1´tpn{k|x0qu ´p1 ` βpx0qq 1 k n ÿ i“1 Khnpx0´ Xiq ˜ Z1´t,i p UZ1´tpn{k|x0q ¸´2α´βpx0q 1ltZ 1´t,ią pUZ1´tpn{k|x0qu `OPpδ1´tqu and

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d2∆pα,1´t1´t|x0q dδ1´t2 “ p1 ` αq # α ż8 1 hα´1py; δ1´t, βq ˆ d dδ1´t hpy; δ1´t, βq ˙2 dy1 k n ÿ i“1 Khnpx0´ Xiq1ltZ1´t,ią pU Z1´tpn{k|x0qu ` ż8 1 hαpy; δ1´t, βq d2 dδ2 1´t hpy; δ1´t, βqdy 1 k n ÿ i“1 Khnpx0´ Xiq1ltZ1´t,ią pU Z1´tpn{k|x0qu ´α ´ 1 k n ÿ i“1 Khnpx0´ Xiqh α´2 ˜ Z1´t,i p UZ1´tpn{k|x0q ; δ1´t, β ¸ ˆ ˜ d dδ1´t h ˜ Z1´t,i p UZ1´tpn{k|x0q ; δ1´t, β ¸¸2 1ltZ 1´t,ią pUZ1´tpn{k|x0qu ´1 k n ÿ i“1 Khnpx0´ Xiqh α´1 ˜ Z1´t,i p UZ1´tpn{k|x0q ; δ1´t, β ¸ ˆ d 2 dδ1´t2 h ˜ Z1´t,i p UZ1´tpn{k|x0q; δ1´t, β ¸ 1ltZ 1´t,ią pUZ1´tpn{k|x0qu + “ p1 ` αq # αβ2px0qp´7 ` βpx0q ` 4α2` 2αβpx0qq p1 ` 2αqp1 ` 2α ` βpx0qqp1 ` 2α ` 2βpx0qq 1 k n ÿ i“1 Khnpx0´ Xiq1ltZ1´t,ią pUZ1´tpn{k|x0qu ´1 ` α k n ÿ i“1 Khnpx0´ Xiq ˜ Z1´t,i p UZ1´tpn{k|x0q ¸´2α 1l tZ1´t,ią pUZ1´tpn{k|x0qu `2p1 ` αqp1 ` βpx0qq 1 k n ÿ i“1 Khnpx0´ Xiq ˜ Z1´t,i p UZ1´tpn{k|x0q ¸´2α´βpx0q 1ltZ 1´t,ią pUZ1´tpn{k|x0qu ´r1 ` α ` 2βpx0q ` 2αβpx0q ` αβ2px0q ´ β2px0qs ˆ1 k n ÿ i“1 Khnpx0´ Xiq ˜ Z1´t,i p UZ1´tpn{k|x0q ¸´2α´2βpx0q 1l tZ1´t,ią pUZ1´tpn{k|x0qu `OPpδ1´tqu .

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In this section we work under the assumptions of Theorem 2.1. b khd n d p∆α,1´tpδ1´t|x0q dδ1´t |δ 1´t“δ1´tp pUZ1´tpn{k|x0q|x0q “ p1 ` αq " 2αβpx0q p1 ` 2αqp1 ` 2α ` βpx0qq ˆb khd n rTn,1´tpsn,1´tp1|x0q|x0q ´ sn,1´tp1|x0qfXpx0qs ´ W1´tpsn,1´tp1|x0qq ˙ ` 2αβpx0q p1 ` 2αqp1 ` 2α ` βpx0qq ˆb khd n rsn,1´tp1|x0q ´ 1s fXpx0q ` W1´tp1q ˙ ` 2αβpx0q p1 ` 2αqp1 ` 2α ` βpx0qqrW1´tpsn,1´tp1|x0qq ´ W1´tp1qs ` b khd n „ Sn,1´tp´2α|x0q ´ 1 1 ` 2αfXpx0q  ´p1 ` βpx0qq b khd n „ Sn,1´tp´2α ´ βpx0q|x0q ´ 1 1 ` 2α ` βpx0q fXpx0q  ` OP ˆb khd nδ1´t ´ p UZ1´tpn{k|x0q ˇ ˇ ˇx0 ¯˙* . Now, remark that by Proposition B.1.10 in de Haan and Ferreira (2006), for n large, with

arbitrary large probability, we have for ε, ξ ą 0 b khd n ˇ ˇ ˇδ1´t ´ p UZ1´tpn{k|x0q ˇ ˇ ˇx0 ¯ˇ ˇ ˇ “ b khd n ˇ ˇ ˇδ1´t ´ UZ1´tpn{k|x0q ˇ ˇ ˇx0 ¯ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ δ1´t ´ p UZ1´tpn{k|x0q ˇ ˇ ˇx0 ¯ δ1´t ´ UZ1´tpn{k|x0q ˇ ˇ ˇx0 ¯ ˇ ˇ ˇ ˇ ˇ ˇ ď b khd n ˇ ˇ ˇδ1´t ´ UZ1´tpn{k|x0q ˇ ˇ ˇx0 ¯ˇ ˇ ˇ » –ε ˜ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q ¸´βpx0q˘ξ ` ˜ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q ¸´βpx0qfi fl “ oPp1q.

Using Lemma 5.2 from Goegebeur et al. (2020), Lemma 4.1 and (15), both combined with the Skorohod construction, and Theorem 2.1, yields

b khd n d p∆α,1´tpδ1´t|x0q dδ1´t |δ 1´t“δ1´tp pUZ1´tpn{k|x0q|x0q ´2αp1 ` αq ż1 0 „ W1´tpzq z ´ W1´tp1q  z2αdz `p1 ` βpx0qqp2α ` βpx0qqp1 ` αq ż1 0 „ W1´tpzq z ´ W1´tp1q  z2α`βpx0qdz. (18)

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Also d2∆pα,1´t1´t|x0q dδ1´t2 |δ1´t“δ1´tp pUZ1´tpn{k|x0q|x0q “ β 2px 0qp1 ` αqp1 ` βpx0q ` 4α2` 2αβpx0qq p1 ` 2αqp1 ` 2α ` βpx0qqp1 ` 2α ` 2βpx0qq fXpx0q `αβ 2px 0qp1 ` αqp´7 ` βpx0q ` 4α2` 2αβpx0qq p1 ` 2αqp1 ` 2α ` βpx0qqp1 ` 2α ` 2βpx0qq rTn,1´tpsn,1´tp1|x0q|x0q ´ sn,1´tp1|x0qfXpx0qs `αβ 2px 0qp1 ` αqp´7 ` βpx0q ` 4α2` 2αβpx0qq p1 ` 2αqp1 ` 2α ` βpx0qqp1 ` 2α ` 2βpx0qq rsn,1´tp1|x0q ´ 1s fXpx0q ´p1 ` αq2 „ Sn,1´tp´2α|x0q ´ fXpx0q 1 ` 2α  `2p1 ` αq2p1 ` βpx0qq „ Sn,1´tp´2α ´ βpx0q|x0q ´ fXpx0q 1 ` 2α ` βpx0q  ´p1 ` αq“1 ` α ` 2βpx0q ` 2αβpx0q ` αβ2px0q ´ β2px0q ‰ „ Sn,1´tp´2α ´ 2βpx0q|x0q ´ fXpx0q 1 ` 2α ` 2βpx0q  `OP ´ δ1´t ´ p UZ1´tpn{k|x0q ˇ ˇ ˇx0 ¯¯ “ β 2 px0qp1 ` αq“1 ` βpx0q ` 4α2` 2αβpx0q ‰ p1 ` 2αqp1 ` 2α ` βpx0qqp1 ` 2α ` 2βpx0qq fXpx0q ` oPp1q. (19) Proof of Theorem 2.2

We limit the proof to deriving the asymptotic properties of a single MDPD estimator pδn,1´t.

The joint asymptotic behavior of ppδn,1´t1, . . . , pδn,1´tJq follows then from Theorem 2.1.

(i) Existence and consistency

The idea is to adjust the arguments used to prove existence and consistency of solutions of the likelihood estimating equations, see, for instance, Theorems 3.7 and 5.1 in Chapter 6 of Lehmann and Casella (1998), to the MDPD framework. To this aim, we start to prove that, for any r ą 0 sufficiently small, we have

P ´ p ∆α,1´t ´ δ1´t ´ p UZ1´tpn{k|x0q ˇ ˇ ˇx0 ¯ˇ ˇ ˇx0 ¯ ă p∆α,1´tpδ1´t|x0q for δ1´t“ δ1´tp pUZ1´tpn{k|x0q|x0q ˘ r ¯ ÝÑ 1,(20)

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p ∆α,1´tpδ1´t|x0q ´ p∆α,1´tpδ1´tp pUZ1´tpn{k|x0q|x0q|x0q “ ´ δ1´t´ δ1´tp pUZ1´tpn{k|x0q|x0q ¯d p∆α,1´t1´t|x0q dδ1´t ˇ ˇ ˇ ˇ ˇ δ1´t“δ1´tp pUZ1´tpn{k|x0q|x0q `1 2 ´ δ1´t´ δ1´tp pUZ1´tpn{k|x0q|x0q ¯2 d2∆pα,1´t1´t|x0q dδ2 1´t ˇ ˇ ˇ ˇ ˇ δ1´t“δ1´tp pUZ1´tpn{k|x0q|x0q `1 6 ´ δ1´t´ δ1´tp pUZ1´tpn{k|x0q|x0q ¯3 d3∆pα,1´t1´t|x0q dδ3 1´t ˇ ˇ ˇ ˇ ˇ δ1´t“rδ1´t “: T1,n` T2,n` T3,n,

where rδ1´tis an intermediate value between δ1´tand δ1´tp pUZ1´tpn{k|x0q|x0q. According to (18),

we have |T1,n| ď r3, whereas according to (19), there exists c ą 0 such that T2,n ą cr2 with

probability tending to 1. Additionally, tedious computations allow us to show that sup δ1´tPrδ1´tp pUZ1´tpn{k|x0q|x0q´r;δ1´tp pUZ1´tpn{k|x0q|x0q`rs ˇ ˇ ˇ ˇ ˇ d3p α,1´tpδ1´t|x0q dδ3 1´t ˇ ˇ ˇ ˇ ˇ ă M,

with arbitrary large probability, from which we can deduce that |T3,n| ď M r3{6 with probability

tending to 1.

Combining all these bounds, we deduce that with probability tending to 1, we have T1,n` T2,n` T3,ną cr2´ p1 ` M {6qr3,

which yields (20).

To complete the proof, we adjust the line of argumentation of Theorem 3.7 in Chapter 6 of Lehmann and Casella (1998). To this aim, for any r ą 0 sufficiently small, we define

Snprq :“ ! p ∆α,1´t ´ δ1´t ´ p UZ1´tpn{k|x0q ˇ ˇ ˇx0 ¯ˇ ˇ ˇx0 ¯ ă p∆α,1´tpδ1´t|x0q for δ1´t“ δ1´tp pUZ1´tpn{k|x0q|x0q ˘ r ) . For v P Snprq, since p∆α,1´tpδ1´t|x0q is differentiable with respect to δ1´t, there exists rδn,1´t P

´

δ1´tp pUZ1´tpn{k|x0q|x0q ´ r; δ1´tp pUZ1´tpn{k|x0q|x0q ` r

¯

where p∆α,1´tpδ1´t|x0q achieves a local

minimum.

According to (20), we have PpSnprqq Ñ 1 for any small enough r, and hence there exists a

sequence rn Ó 0 such that PpSnprnqq Ñ 1 as n Ñ 8. Now, let pδn,1´t:“ rδn,1´t if v P Snprnq and

arbitrary otherwise. Since v P Snprnq implies

d p∆α,1´tpδ1´t|x0q dδ1´t ˇ ˇ ˇ ˇ ˇ δ1´t“pδn,1´t “ 0, we have P ¨ ˝ d p∆α,1´tpδ1´t|x0q dδ1´t ˇ ˇ ˇ ˇ ˇ δ1´t“pδn,1´t “ 0 ˛ ‚ě PpSnprnqq ÝÑ 1,

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as n Ñ 8, which establishes the existence part.

Concerning now the consistency part, note that for any r ą 0 and n large enough such that rnď r, we have P ´ˇ ˇ ˇpδn,1´t´ δ1´tp pUZ1´tpn{k|x0q|x0q ˇ ˇ ˇ ă r ¯ ě P ´ˇ ˇ ˇδpn,1´t´ δ1´tp pUZ1´tpn{k|x0q|x0q ˇ ˇ ˇ ă rn ¯ ě PpSnprnqq ÝÑ 1,

as n Ñ 8, whence the consistency of the estimator sequence.

(ii) Asymptotic normality

By definition we have d p∆α,1´tpδ1´t|x0q dδ1´t ˇ ˇ ˇ δ1´t“pδn,1´t

“ 0. Thus a Taylor series expansion around

δ1´tp pUZ1´tpn{k|x0q|x0q combined with the boundedness of the third derivative of p∆α,1´tpδ1´t|x0q

with respect to δ1´t leads to

0 “ d p∆α,1´tpδ1´t|x0q dδ1´t ˇ ˇ ˇ ˇ ˇ δ1´t“δ1´tp pUZ1´tpn{k|x0q|x0q `d 2 p ∆α,1´tpδ1´t|x0q dδ21´t ˇ ˇ ˇ ˇ ˇ δ1´t“δ1´tp pUZ1´tpn{k|x0q|x0q ´ p δn,1´t´ δ1´tp pUZ1´tpn{k|x0q|x0q ¯ p1 ` oPp1qq

from which we deduce that

b khd n ´ p δn,1´t´ δ1´tp pUZ1´tpn{k|x0q|x0q ¯ “ ´ ¨ ˝ d2∆pα,1´t1´t|x0q dδ2 1´t ˇ ˇ ˇ ˇ ˇ δ1´t“δ1´tp pUZ1´tpn{k|x0q|x0q ˛ ‚ ´1 b khd n d p∆α,1´tpδ1´t|x0q dδ1´t ˇ ˇ ˇ ˇ ˇ δ1´t“δ1´tp pUZ1´tpn{k|x0q|x0q p1 ` oPp1qq c " 2α ż1 0 „ W1´tpzq z ´ W1´tp1q  z2αdz ´ p1 ` βpx0qqp2α ` βpx0qq ż1 0 „ W1´tpzq z ´ W1´tp1q  z2α`βpx0qdz * .  Proof of Theorem 2.3

Again we limit the proof to showing the result for a single estimator pLkpy1, y2|x0q. The finite

dimensional convergence follows then from Theorem 2.2 and Lemma 4.3. From (3) and (4), we deduce that

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b khd n ´ p Lkpy1, y2|x0q ´ Lpy1, y2|x0q ¯ “ ´y1 b khd n ´ p K1´t,kpx0q ´ K1´tpx0q ¯ “ ´y1 b khd n ˜ k n p UZ1´tpn{k|x0q 1 ` pδn,1´t ´ K1´tpx0q ¸ “ ´y1K1´tpx0q b khd n ˜ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q 1 ` δ1´tpUZ1´tpn{k|x0q|x0q 1 ` pδn,1´t ´ 1 ¸ “ ´y1K1´tpx0q b khd n ˜ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q ´ 1 ¸ `y1K1´tpx0q b khd n ´ p δn,1´t´ δ1´tpUZ1´tpn{k|x0q|x0q ¯ 1 1 ` pδn,1´t `y1K1´tpx0q p δn,1´t´ δ1´tpUZ1´tpn{k|x0q|x0q 1 ` pδn,1´t b khd n ˜ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q ´ 1 ¸ .

Now remark that b khd n ˇ ˇ ˇδ1´tp pUZ1´tpn{k|x0q|x0q ´ δ1´tpUZ1´tpn{k|x0q|x0q ˇ ˇ ˇ “ b khd n ˇ ˇδ1´tpUZ1´tpn{k|x0q|x0q ˇ ˇ ˇ ˇ ˇ ˇ ˇ δ1´tp pUZ1´tpn{k|x0q|x0q δ1´tpUZ1´tpn{k|x0q|x0q ´ 1 ˇ ˇ ˇ ˇ ˇ “ oPp1q,

by (14). This implies that b khd n ´ p Lkpy1, y2|x0q ´ Lpy1, y2|x0q ¯ “ ´y1K1´tpx0q b khd n ˜ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q ´ 1 ¸ `y1K1´tpx0q b khd n ´ p δn,1´t´ δ1´tp pUZ1´tpn{k|x0q|x0q ¯ ` oPp1q.

Using the fact that

b khd n ¨ ˚ ˚ ˝ p UZ1´tpn{k|x0q UZ1´tpn{k|x0q´ 1 p δn,1´t´ δ1´tp pUZ1´tpn{k|x0q|x0q ˛ ‹ ‹ ‚ ¨ ˚ ˝ W1´tp1q fXpx0q c ´ 2αş10 ” W1´tpzq z ´ W1´tp1q ı z2αdz ´ p1 ` βpx 0qqp2α ` βpx0qq ş1 0 ” W1´tpzq z ´ W1´tp1q ı z2α`βpx0qdz¯ ˛ ‹ ‚,

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b khd n ´ p Lkpy1, y2|x0q ´ Lpy1, y2|x0q ¯ ´y1K1´tpx0q W1´tp1q fXpx0q ` y1K1´tpx0qc " 2α ż1 0 „ W1´tpzq z ´ W1´tp1q  z2αdz ´p1 ` βpx0qqp2α ` βpx0qq ż1 0 „ W1´tpzq z ´ W1´tp1q  z2α`βpx0qdz * . 

4.5 Auxiliary results in case of unknown margins

In order to establish the weak convergence of statistics qSn,1´tps|x0q, we need, as a preliminary

result, the weak convergence of statistics qTn,1´tpy|x0q. This is the aim of the next proposition.

Let | Wn,1´tj :“ "b khd n ´ q Tn,1´tjpy|x0q ´ yfXpx0q ¯ ; y P p0, T s * , for j “ 1, . . . , J .

Proposition 4.1 Assume that there exists b ą 0 such that fXpxq ě b, @x P SX Ă Rd, fX

is bounded, pD1´tjq, pH1´tjq for j “ 1, . . . , J , pD0.5q, pH0.5q, pK2q, pFmq hold, and that y Ñ

FZ1´tjpy|x0q, j “ 1, . . . , J , are strictly increasing at x0 P IntpSXq non-empty. Consider

se-quences k Ñ 8, hnÑ 0 and cnÑ 0 as n Ñ 8, such that k{n Ñ 0, khdnÑ 8, h

ηε1´t1^¨¨¨^ηε1´t J^ηε0.5 n lognk Ñ 0, akhd nh ηfXK1´t1^¨¨¨^ηK1´t J n Ñ 0, a khd n|δ1´tpUZ1´tjpnk|x0q|x0q|h ηC1´t j n Ñ 0, j “ 1, . . . , J, a khd n|δ1´tjpUZ1´tjp n k|x0q|x0q|h ηε1´t j

n lognk Ñ 0, j “ 1, . . . , J . Under conditions (7) and (8), we

have ´ | Wn,1´t1, . . . , |Wn,1´tJ ¯ pW1´t1, . . . , W1´tJq , in `Jpp0, T sq, for any T ą 0.

Then, the next ingredients required in order to proof the weak convergence of qSn,1´tps|x0q are

lemmas similar to Lemma 4.3 fom the present paper and Lemma 5.2 in Goegebeur et al. (2020), but this time for the variable qZ1´t instead of Z1´t. This is the aim of Lemmas 4.5 and 4.6 below

for which we need first to show the weak convergence of pFZq

1´tp.|x0q correctly normalized.

Lemma 4.4 Under the assumptions of Proposition 4.1, for any sequence upjqn satisfying

b khd n ¨ ˝ FZ1´tjpUZ1´tjpn{k|x0q|x0q FZ1´tjpupjqn |x0q ´ 1 ˛ ‚Ñ cj P R,

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as n Ñ 8, j “ 1, . . . , J , we have ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˝ b nhd nFZ1´t1pu p1q n |x0q ˜ p F q Z1´t1pu p1q n |x0q FZ1´t1pup1qn |x0q ´ 1 ¸ .. . b nhd nFZ1´tJpupJ qn |x0q ˜ p FZ1´tq J pupJ qn |x0q FZ1´t Jpu pJ q n |x0q´ 1 ¸ ˛ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‚ 1 fXpx0q ¨ ˚ ˝ W1´t1p1q .. . W1´tJp1q ˛ ‹ ‚.

Lemma 4.5 Assume that there exists b ą 0 such that fXpxq ě b, @x P SX Ă Rd, fX is bounded,

pD1´tjq, pH1´tjq, for j “ 1, . . . , J , pD0.5q, pH0.5q, pK2q, pFmq hold, and that y Ñ FZ1´tjpy|x0q,

j “ 1, . . . , J , are strictly increasing at x0 P IntpSXq non-empty. Consider sequences k Ñ 8,

hn Ñ 0 and cnÑ 0 as n Ñ 8, such that k{n Ñ 0, khdnÑ 8, h

ηε1´t1^¨¨¨^ηε1´t J^ηε0.5 n lognk Ñ 0, a khd nh ηfXK1´t1^¨¨¨^ηK1´t J n Ñ 0, a khd n|δ1´tjpUZ1´tjp n k|x0q|x0q| Ñ 0, j “ 1, . . . , J . Under

conditions (7) and (8), we have

b khd n ¨ ˚ ˚ ˚ ˚ ˚ ˝ p U q Z1´t1pn{k|x0q UZ1´t1pn{k|x0q´ 1 .. . p U q Z1´tJ pn{k|x0q UZ1´t Jpn{k|x0q ´ 1 ˛ ‹ ‹ ‹ ‹ ‹ ‚ 1 fXpx0q ¨ ˚ ˝ W1´t1p1q .. . W1´tJp1q ˛ ‹ ‚.

From Lemma 4.5, we can show now the uniform convergence in probability of qsn,1´tpz|x0q towards z for any z P p0, T s.

Lemma 4.6 Under the assumptions of Lemma 4.5, for any T ą 0, we have sup

zPp0,T s

|qsn,1´tpz|x0q ´ z| “ oPp1q.

4.6 Proof of the auxiliary results in case of unknown margins

Proof of Proposition 4.1. Firstly, we consider the weak convergence of a single process takhd

np qTn,1´tpy|x0q ´ yfXpx0qq; y P p0, T su, where for simplicity of notation we have ignored the

index j from t. We use the decomposition b khd n ´ q Tn,1´tpy|x0q ´ yfXpx0q ¯ “ b khd npTn,1´tpy|x0q ´ yfXpx0qq ` b khd n ´ q Tn,1´tpy|x0q ´ Tn,1´tpy|x0q ´ E ” q Tn,1´tpy|x0q ´ Tn,1´tpy|x0q ı¯ ` b khd nE ” q Tn,1´tpy|x0q ´ Tn,1´tpy|x0q ı “: 3 ÿ i“1 Qi,1´tpy|x0q. (21)

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According to Lemma 4.1, we have

Q1,1´tpy|x0q W1´tpyq, (22)

in `pp0, T sq.

The next step consists in showing that sup

yPp0,T s

|Q2,1´tpy|x0q| “ oPp1q. (23)

To this aim, we will make use of empirical process theory with changing function classes, see for instance van der Vaart and Wellner (1996). We start by introducing some notation. Let P be the distribution measure of pYp1q, Yp2q, Xq, and denote the expected value under P as

P f :“ş fdP for any real-valued measurable function f : R2ˆ RdÑ R. For a function class F , define now the covering number N pF , L2pQq, τ q as the minimal number of L2pQq-balls of radius

τ needed to cover the class of functions F and the uniform entropy integral as J pδ, F , L2q :“ żδ 0 c log sup QPQ N pF , L2pQq, τ }F }Q,2q dτ,

where Q is the set of all probability measures Q for which 0 ă }F }2Q,2:“ş F2dQ ă 8 and F is an envelope function for the class F .

Let In:“ tgy,δ,n: y P p0, T s, δ P Hu where H :“ δ “ pδ1, δ2q; δ : R ˆ R ˆ SX Ñ R2( , and gy,δ,npv1, v2, uq :“ c nhd n

k Khnpx0´ uqqy,δ,npv1, v2, uq,

qy,δ,npv1, v2, uq :“ 1ltF Z1´tpZδpv1,v2,uq|x0qănkyu, with Zδpv1, v2, uq :“ min ˆ 1 |1 ´ δ1pv1, v2, uq| ,1 ´ t t 1 |1 ´ δ2pv1, v2, uq| ˙ .

For convenience, denote δn :“

´ p

Fn,1, pFn,2

¯

and δ0 :“ pF1, F2q. According to Lemma 3.1 in

Escobar-Bach et al. (2018a), if rn :“ maxp

a

| log cn|q{ncdn, c η

nq, we have r´1n |δn´ δ0| converges

in probability towards the null function H0 :“ t0u in H, endowed with the norm }δ}H :“

}δ1}8` }δ2}8 for any δ P H. We want to apply Theorem 2.3 in Van der Vaart and Wellner

(2007). To this aim, we consider the class

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with envelope function given by Gnpy, bqpv1, v2, uq :“ c nhd n k Khnpx0´ uq1l" 1

UZ1´tpn{pkyq|x0qPr1´F1pv1|uq´rnb;1´F1pv1|uq`rnbs

* ` c nhd n k Khnpx0´ uq1l" 1 UZ1´tpn{pkyq|x0qPr t 1´tp1´F2pv2|uq´rnbq; t 1´tp1´F2pv2|uq`rnbqs * since

|gy,δ0`rnδ,npv1, v2, uq ´ gy,δ0,npv1, v2, uq|

“ c

nhd n

k Khnpx0´ uq|qy,δ0`rnδ,npv1, v2, uq ´ qy,δ0,npv1, v2, uq|

“ c nhd n k Khnpx0´ uq ˇ ˇ

ˇ1ltZδ0`rnδpv1,v2,uqąUZ1´tpn{pkyq|x0qu´ 1ltZδ0pv1,v2,uqąUZ1´tpn{pkyq|x0qu

ˇ ˇ ˇ ď c nhd n

k Khnpx0´ uq1ltUZ1´tpn{pkyq|x0qPrminpZδ0`rnδpv1,v2,uq,Zδ0pv1,v2,uqq,maxpZδ0`rnδpv1,v2,uq,Zδ0pv1,v2,uqqsu

ď c nhd n k Khnpx0´ uq ˆ " 1lt 1

UZ1´tpn{pkyq|x0qPrminp|1´F1pv1|uq´rnδ1|,1´F1pv1|uqq,maxp|1´F1pv1|uq´rnδ1|,1´F1pv1|uqqs

`1l"

1

UZ1´tpn{pkyq|x0qPr t

1´tminp|1´F2pv2|uq´rnδ2|,1´F2pv2|uqq, t

1´tmaxp|1´F2pv2|uq´rnδ2|,1´F2pv2|uqqs

* , . -ď Gnpy, bqpv1, v2, uq. We need to show

Assertion 1: supyPp0,T s?nP Gnpy, bnq ÝÑ 0 for every bnÑ 0

and

Assertion 2: Let Gn be an envelope for tGnpy, bq : y P p0, T su, we have

A) P G2n“ Op1q; B) P G2ntGně ε

?

nu Ñ 0, @ε ą 0; C) supyPp0,T sP G2npy, bq Ñ 0;

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Proof of Assertion 1. Remark that ? nP Gnpy, bnq “ n c hd n k $ & % E » –Khnpx0´ XqE $ & % 1l" 1 UZ1´tpn{pkyq|x0qPr1´F1pYp1q|Xq´rnbn,1´F1pYp1q|Xq`rnbns * ˇ ˇ ˇX , . -fi fl `E » –Khnpx0´ XqE $ & % 1l" 1 UZ1´tpn{pkyq|x0qPr t 1´tp1´F2pYp2q|Xq´rnbnq, t 1´tp1´F2pYp2q|Xq`rnbnqs * ˇ ˇ ˇX , . -fi fl , . -ď n c hd n k 4rnbnE rKhnpx0´ Xqs .

Thus Assertion 1 is satisfied as soon as rnn

b

hd n

k Ñ 0.

Proof of Assertion 2 A). Define

Gp1qn pbqpv1, uq :“ c nhd n k Khnpx0´ uq1l" 1´F1pv1|uqďUZ 1 1´tpn{pkT q|x0q `rnb *, Gp2qn pbqpv2, uq :“ c nhd n k Khnpx0´ uq1l" 1´F2pv2|uqď1´tt UZ 1 1´tpn{pkT q|x0q `rnb *, and set Gnpbqpv1, v2, uq :“ G p1q n pbqpv1, uq ` G p2q

n pbqpv2, uq. This yields

P G2n ď 2nh d n k E " Kh2npx0´ XqP ˆ 1 ´ F1pYp1q|Xq ď 1 UZ1´tpn{pkT q|x0q ` rnb ˇ ˇ ˇX ˙* `2nh d n k E " Kh2npx0´ XqP ˆ 1 ´ F2pYp2q|Xq ď 1 ´ t t 1 UZ1´tpn{pkT q|x0q ` rnb ˇ ˇ ˇX ˙* ď C„ n k 1 UZ1´tpn{pkT q|x0q `n krnb  “ Op1q, since khdnÑ 8 and rnn b hd n

k Ñ 0. This achieves the proof of Assertion 2 A).

Proof of Assertion 2 B). Clearly, we have G2`αn ď C „ ´ Gp1qn ¯2`α ` ´ Gp2qn ¯2`α .

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Thus @ε ą 0, we have P G2ntGně ε ? nu ď 1 nα{2εαE ´ G2`αn ¯ ď C εα n k 1 pkhdnqα{2 ˆ " E„ 1 hd n K2`αˆ x0´ X hn ˙ P ˆ 1 ´ F1pYp1q|Xq ď 1 UZ1´tpn{pkT q|x0q ` rnb ˇ ˇ ˇX ˙ ` E„ 1 hd n K2`αˆ x0´ X hn ˙ P ˆ 1 ´ F2pYp2q|Xq ď 1 ´ t t 1 UZ1´tpn{pkT q|x0q ` rnb ˇ ˇ ˇX ˙* ď C εα n k 1 pkhdnqα{2 „ 1 UZ1´tpn{pkT q|x0q ` rnb 

which tends again to 0 under our assumptions.

Proof of Assertion 2 C). Following the lines of proof of Assertion 1, we have sup

yPp0,T s

P G2npy, bq ď Crn

n k Ñ 0.

Proof of Assertion 2 D). We follow the lines of proof of Theorem 2.2 in Escobar-Bach et al. (2018b). The class of functions on r0, 1s

u Ñ 1lty1ď1´uďy2u, y1ă y2( ,

is a V C-class. This allows us to prove that there exist positive constants C and V such that sup QPQ N ptGnpy, bq : y P p0, T su , L2pQq, τ }Gn}Q,2q ď C ˆ 1 τ ˙V , from which Assertion 2 D) follows. This achieves the proof of (23).

It remains to show that

sup yPp0,T s |Q3,1´tpy|x0q| “ op1q. (24) Note that b khd n ˇ ˇ ˇE ” q Tn,1´tpy|x0q ´ Tn,1´tpy|x0q ıˇ ˇ ˇ “ n c hd n k ˇ ˇ ˇE ” Khnpx0´ Xq ´ 1lt1´F Z1´tp qZ1´t|x0qăknyu´ 1lt1´FZ1´tpZ1´t|x0qănkyu ¯ıˇ ˇ ˇ ď ?n c nhd n k E ” Khnpx0´ Xq ˇ ˇ ˇ1lt1´F Z1´tp qZ1´t|x0qăknyu´ 1lt1´FZ1´tpZ1´t|x0qăknyu ˇ ˇ ˇ ı “ ?n c nhd n k E ” Khnpx0´ Xq ˇ ˇ ˇ1lt1´FZ1´tpZδn|x0qănkyu´ 1lt1´FZ1´tpZδ0|x0qăknyu ˇ ˇ ˇ ı ď ?nP Gnpy, bq,

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for n large enough. This implies that sup yPp0,T s b khd n ˇ ˇ ˇE ” q Tn,1´tpy|x0q ´ Tn,1´tpy|x0q ıˇ ˇ ˇ ď sup yPp0,T s ? nP Gnpy, bq ÝÑ 0

by Assertion 1 since it is clear that bn Ñ 0 can be replaced by any fixed value b as soon as

rnn

b

hd n

k Ñ 0 holds.

Combining (22), (23) and (24) we have shown that b khd n ´ q Tn,1´tpy|x0q ´ yfXpx0q ¯ W1´tpyq, in `pp0, T sq.

Secondly, to obtain the joint weak convergence of p|Wn,1´t1, . . . , |Wn,1´tJq we need to establish

the finite dimensional weak convergence together with tightness. The finite dimensional weak convergence can be shown by the Cram´er-Wold device, as in the proof of Lemma 4.1 along with decomposition (21). The joint tightness follows from the individual tightness (similarly to

Lemma 1 in Bai and Taqqu, 2013). 

Proof of Lemma 4.4. This proof is omitted since it is similar to the proof of Lemma 4.2, by using Proposition 4.1 instead of Lemma 4.1 combining with the Skorohod construction (keeping

the same notation). 

Proof of Lemma 4.5. Again, this proof is omitted since it is similar to the proof of Lemma

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Proof of Lemma 4.6. By definition, we have |qsn,1´tpz|x0q ´ z| “ ˇ ˇ ˇ ˇ ˇ ˇ FZ1´t ´ z´1Up q Z1´tpn{k|x0q ˇ ˇ ˇx0 ¯ FZ1´t`UZ1´tpn{k|x0q|x0 ˘ ´ z ˇ ˇ ˇ ˇ ˇ ˇ “ ˇ ˇ ˇ ˇ ˇ ˇ z ˜ p UZq 1´tpn{k|x0q UZ1´tpn{k|x0q ¸´1 1 ` δ1´t ´ z´1Up q Z1´tpn{k|x0q ˇ ˇ ˇx0 ¯ 1 ` δ1´t`UZ1´tpn{k|x0q|x0 ˘ ´ z ˇ ˇ ˇ ˇ ˇ ˇ ď z ˇ ˇ ˇ ˇ ˇ ˇ ˜ p U q Z1´tpn{k|x0q UZ1´tpn{k|x0q ¸´1 ´ 1 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 1 ` δ1´t ´ z´1Up q Z1´tpn{k|x0q ˇ ˇ ˇx0 ¯ 1 ` δ1´t`UZ1´tpn{k|x0q|x0 ˘ ˇ ˇ ˇ ˇ ˇ ˇ `z ˇ ˇ ˇ ˇ ˇ ˇ 1 ` δ1´t ´ z´1 p UZq 1´tpn{k|x0q ˇ ˇ ˇx0 ¯ 1 ` δ1´t`UZ1´tpn{k|x0q|x0 ˘ ´ 1 ˇ ˇ ˇ ˇ ˇ ˇ ď z ˇ ˇ ˇ ˇ ˇ ˇ ˜ p UZq 1´tpn{k|x0q UZ1´tpn{k|x0q ¸´1 ´ 1 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 1 ` δ1´t ´ z´1 p UZq 1´tpn{k|x0q|x0 ¯ 1 ` δ1´t`UZ1´tpn{k|x0q|x0 ˘ ˇ ˇ ˇ ˇ ˇ ˇ `z |δ1´t`UZ1´tpn{k|x0q|x0 ˘ | |1 ` δ1´t`UZ1´tpn{k|x0q|x0 ˘ | $ & % ˇ ˇ ˇ ˇ ˇ ˇ δ1´t ´ z´1 p UZq 1´tpn{k|x0q|x0 ¯ δ1´t`UZ1´tpn{k|x0q|x0 ˘ ´ ˜ z´1 p UZq 1´tpn{k|x0q UZ1´tpn{k|x0q ¸´βpx0qˇ ˇ ˇ ˇ ˇ ˇ ` ˇ ˇ ˇ ˇ ˇ ˇ ˜ z´1UpZq1´tpn{k|x0q UZ1´tpn{k|x0q ¸´βpx0q ´ 1 ˇ ˇ ˇ ˇ ˇ ˇ , . -.

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proba-bility, we have for ε, ξ ą 0 and z ď T |qsn,1´tpz|x0q ´ z| ď C z ˇ ˇ ˇ ˇ ˇ ˇ ˜ p UZq 1´tpn{k|x0q UZ1´tpn{k|x0q ¸´1 ´ 1 ˇ ˇ ˇ ˇ ˇ ˇ `C ε zˇˇδ1´t`UZ1´tpn{k|x0q|x0 ˘ˇ ˇ ˜ z´1 p UZq 1´tpn{k|x0q UZ1´tpn{k|x0q ¸´βpx0q˘ξ `C zˇˇδ1´t`UZ1´tpn{k|x0q|x0 ˘ˇ ˇ $ & % zβpx0q ˜ p UZq 1´tpn{k|x0q UZ1´tpn{k|x0q ¸´βpx0q ` 1 , . -ď C T ˇ ˇ ˇ ˇ ˇ ˇ ˜ p UZq 1´tpn{k|x0q UZ1´tpn{k|x0q ¸´1 ´ 1 ˇ ˇ ˇ ˇ ˇ ˇ `C ε T1`βpx0q˘ξˇ ˇδ1´t`UZ 1´tpn{k|x0q|x0 ˘ˇ ˇ ˜ p UZq 1´tpn{k|x0q UZ1´tpn{k|x0q ¸´βpx0q˘ξ `C Tˇˇδ1´t`UZ 1´tpn{k|x0q|x0 ˘ˇ ˇ $ & % Tβpx0q ˜ p UZq 1´tpn{k|x0q UZ1´tpn{k|x0q ¸´βpx0q ` 1 , . -.

Using Lemma 4.5, Lemma 4.6 follows. 

4.7 Proof of Theorem 3.1.

This proof is omitted since it follows exactly the same lines of proof as those for Theorem 2.1

but with the auxiliary results in case of unknown margins. 

4.8 Proof of Theorem 3.2.

This proof is omitted since it follows exactly the same lines of proof as those for Theorem 2.3

with the auxiliary results in case of unknown margins. 

Acknowledgement

The research of Armelle Guillou was supported by the French National Research Agency under the grant ANR-19-CE40-0013-01/ExtremReg project and an International Emerging Action (IEA-00179). Computation/simulation for the work described in this paper was supported by the DeIC National HPC Centre, SDU.

References

Bai, S., Taqqu, M.S., 2013. Multivariate limit theorems in the context of long-range dependence. Journal of Time Series Analysis, 34, 717–743.

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Basu, A., Harris, I.R., Hjort, N.L., Jones, M.C., 1998. Robust and efficient estimation by mini-mizing a density power divergence. Biometrika, 85, 549–559.

Beirlant, J., Joossens, E., Segers, J., 2009. Second-order refined peaks-over-threshold modelling for heavy-tailed distributions. Journal of Statistical Planning and Inference, 139, 2800–2815. Billingsley, P., 1995. Probability and Measure, third edition. Wiley.

Dutang, C., Goegebeur, Y., Guillou, A., 2014. Robust and bias-corrected estimation of the coefficient of tail dependence. Insurance: Mathematics and Economics, 57, 46–57.

Escobar-Bach, M., Goegebeur, Y., Guillou, A., 2018a. Local robust estimation of the Pickands dependence function. Annals of Statistics, 46, 2806–2843.

Escobar-Bach, M., Goegebeur, Y., Guillou, A., 2018b. Local estimation of the conditional stable tail dependence function. Scandinavian Journal of Statistics, 45, 590–617.

Escobar-Bach, M., Goegebeur, Y., Guillou, A., 2020. Bias correction in conditional multivariate extremes. Electronic Journal of Statistics, 14, 1773–1795.

Escobar-Bach, M., Goegebeur, Y., Guillou, A., You, A., 2017. Bias-corrected and robust esti-mation of the bivariate stable tail dependence function. TEST, 26, 284–307.

Feuerverger, A., Hall, P., 1999. Estimating a tail exponent by modelling departure from a Pareto distribution. Annals of Statistics, 27, 760–781.

Gardes, L., Girard, S., 2015. Nonparametric estimation of the conditional tail copula. Journal of Multivariate Analysis, 137, 1–16.

Gin´e, E., Guillou, A., 2002. Rates of strong uniform consistency for multivariate kernel density estimators. Annales de l’Institut Henri Poincar´e, Probabilit´es et Statistiques, 38, 907–921. Gin´e, E., Koltchinskii, V., Zinn, J., 2004. Weighted uniform consistency of kernel density esti-mators. Annals of Probability, 32, 2570–2605.

Goegebeur, Y., Guillou, A., Ho, N.K.L., Qin, J., 2020. A Weissman-type estimator of the condi-tional marginal expected shortfall. Available at https://hal.archives-ouvertes.fr/hal-02613135v2/. Goegebeur, Y., Guillou, A., Ho, N.K.L., Qin, J., 2021. Robust nonparametric estimation of the conditional tail dependence coefficient. Available at https://doi.org/10.1016/j.jmva.2020.104607. Gomes, M.I., Martins, M.J., 2004. Bias-reduction and explicit semi-parametric estimation of the tail index. Journal of Statistical Planning and Inference, 124, 361–378.

(40)

de Haan, L., Ferreira, A., 2006. Extreme Value Theory - An Introduction. Springer.

Huang, X., 1992. Statistics of bivariate extremes. PhD Thesis, Erasmus University Rotterdam, Tinbergen Institute Research series No. 22.

Lehmann, E.L., Casella, G. 1998. Theory of Point Estimation, second edition, Springer, New York.

Mhalla, L., de Carvalho, M., Chavez-Demoulin, V., 2019. Regression type models for extremal dependence. Scandinavian Journal of Statistics, 46, 1141–1167.

Nolan, D., Pollard, D., 1987. U-processes: rates of convergence. Annals of Statistics, 15, 780– 799.

van der Vaart, A. W., Wellner, J. A., 1996. Weak convergence and empirical processes, with applications to statistics. Springer Series in Statistics. Springer-Verlag, New York.

van der Vaart, A. W., Wellner, J. A., 2007. Empirical processes indexed by estimated functions. Asymptotics: Particles, Processes and Inverse Problems. IMS Lecture Notes Monogr. Ser. 55, 234–252.

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