Yuri Goegebeur

Armelle Guillou

Nguyen Khanh

Le Ho

Jing Qin

Nonparametric estimation of conditional marginal excess moments

Several risk measures have been proposed in the literature, among them the marginal mean excess, defined as M M E p " ErpY p1q ´Q1 p1 ´pqq `|Y p2q ą Q 2 p1 ´pqs, provided E|Y p1q | ă 8, where pY p1q , Y p2q q denotes a pair of risk factors, y `:" maxp0, yq, Q j the quantile function of Y pjq , j " 1, 2, and p P p0, 1q. In this paper we consider a generalization of this measure, where the random variables of main interest pY p1q , Y p2q q are observed together with a random covariate X P R d , and where the Y p1q excess is also power transformed. This leads to the concept of conditional marginal excess moment for which an estimator is proposed allowing extrapolation outside the data range. The main asymptotic properties of this estimator have been established, using empirical processes arguments combined with the multivariate extreme value theory. The finite sample behavior of the estimator is evaluated by a simulation experiment. We apply also our method on a vehicle insurance customer dataset.

Introduction

In many scientific disciplines, quantifying risks related to extreme events is of crucial importance. For instance, insurance companies are once in a while faced with extreme claims which can jeopardise the solvency of a portfolio, and hence accurate modelling of the upper tail of the claim size distribution assumes a central place in their risk management. Other examples of disciplines include environmental science (storms, rainfall), geology (severe earthquakes), hydrology (flooding) and telecommunication (network load). The quantification of the risk of a risk factor Y with distribution function F Y is done by so-called risk measures. In the univariate context, the most commonly used risk measures are the Value-at-Risk (VaR), defined as V aR p " Qp1 ´pq, where Q denotes the quantile function of Y , i.e., Qppq :" infty : F Y pyq ě pu, p P p0, 1q, see, e.g., [START_REF] Jorion | Value at risk: the new benchmark for managing financial risk[END_REF] for a review, and the conditional tail expectation (CTE), given by CT E p " EpY |Y ą Qp1 ´pqq, p P p0, 1q, provided E|Y | ă 8. During recent years, the CTE became a popular alternative to VaR since, compared to VaR it is more conservative and it is also a coherent risk measure. We refer to, e.g., [START_REF] Artzner | Coherent measures of risk[END_REF], [START_REF] Cai | Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures[END_REF] and [START_REF] Brazaukas | Estimating conditional tail expectation with actuarial applications in view[END_REF]. In practice, risk is often related to several risk factors, and hence the above discussed risk measures need to be adjusted for this multivariate context. For a pair of risk factors pY p1q , Y p2q q, with E|Y p1q | ă 8, the CTE can be generalized to the marginal expected shortfall (MES), defined as M ES p " EpY p1q |Y p2q ą Q 2 p1 ´pqq, p P p0, 1q,

where Q 2 denotes the quantile function of the risk factor Y p2q . This measure was introduced by Acharya et al. (2010), to measure the contribution of a financial firm to an overall systemic risk. For a financial firm, the MES is defined as its short-run expected equity loss conditional on the market taking a loss greater than its VaR. [START_REF] Cai | Estimation of the marginal expected shortfall: the mean when a related variable is extreme[END_REF] studied the MES in a bivariate extreme value framework, and proposed an estimator for it when the Y p2q quantile is extreme, i.e., when p ă 1{n. See also Di Bernardino and Prieur (2018), and Das andFasen-Hartmann (2018, 2019) for related analyses of the MES in a multivariate extreme value setting. By replacing Y p1q in (1) by an excess over a high quantile of its distribution one obtains the marginal mean excess (MME):

M M E p :" ErpY p1q ´Q1 p1 ´pqq `|Y p2q ą Q 2 p1 ´pqs, p P p0, 1q, provided E|Y p1q | ă 8, and where y `:" maxp0, yq and Q 1 is the quantile function of Y p1q . Das andFasen-Hartmann (2018, 2019) study a slightly different version of MME under multivariate regular variation, introduced an estimator for it based on extreme value arguments, and established its consistency.

Recently, several of the above mentioned risk measures have been generalized to the situation where the variable(s) of main interest is (are) observed together with a random covariate. [START_REF] Daouia | Kernel estimators of extreme level curves[END_REF][START_REF] Daouia | On kernel smoothing for extremal quantile regression[END_REF] studied the estimation of extreme conditional quantiles, while the CTE was extended to the regression case in El Methni et al. (2014,2018). Goegebeur et al. (2021a, b) considered the estimation of the MES in presence of random covariates.

In this paper we consider a generalization of the marginal mean excess, where the random variables of main interest pY p1q , Y p2q q are observed together with a random covariate X P R d , and where the Y p1q excess is also power transformed. We assume that the covariate X has a density function f X , with support S X Ă R d , and we denote by F j p.|xq the conditional distribution function of Y pjq given X " x and by U j p.|xq the associated conditional tail quantile function, i.e., U j p.|xq :" infty : F j py|xq ě 1 ´1{.u, j " 1, 2. In particular, we introduce the conditional marginal excess moment (CMEM), defined as θ β,p px 0 q " E " pY p1q ´U1 p1{p|x 0 qq β `ˇY p2q ą U 2 p1{p|x 0 q, X "

x 0 ı , (2)
where β ą 0, provided Ep|Y p1q | β |X " x 0 q ă 8, and where x 0 is a reference position such that x 0 P IntpS X q, the interior of the support of f X , assumed to be non-empty. The motivation for introducing this power β is that, in an insurance context for instance, with Y p1q denoting the claim size, pY p1q ´U1 p1{p|x 0 qq `can be viewed as the payment by the reinsurer. Thus, different values of β allow us to compute, among others, the expectation or the variance of this payment. We are interested in the situation where p is small, i.e., p ă 1{n, with n denoting the sample size. To obtain this extrapolation, we will work in two steps, where in a first step we study an intermediate case, which allows for constructing an empirical estimator for [START_REF] Artzner | Coherent measures of risk[END_REF]. In a second step this intermediate estimator will be extrapolated outside the data range by a Weissman-type construction.

The remainder of the paper is organized as follows. In Section 2 we consider the CMEM in the intermediate case. We introduce a locally weighted average as estimator for the CMEM and derive its limiting distribution under suitable conditions. This intermediate estimator is then extrapolated outside the data range in Section 3, where we study again the asymptotic properties. In Section 4 we evaluate the finite sample performance with a simulation experiment, while in Section 5 we illustrate the method on a vehicle insurance customer dataset. Some auxiliary results and their proofs are given in Section 6. Section 7 contains the proofs of the main results.

2 Estimator for the intermediate case

Let pY p1q i , Y p2q
i , X i q, i " 1, . . . , n, be independent copies of pY p1q , Y p2q , Xq. We start with considering an estimator for θ β,p px 0 q when p Ó 0 at an intermediate rate, i.e., when p " k{n, where k Ñ 8 as n Ñ 8 but in such a way that k{n Ñ 0. In this case it is natural to consider

p θ n :" 1 k n ÿ i"1 K hn px 0 ´Xi q " Y p1q i ´p U 1 ´n k |x 0 ¯ıβ `1l tY p2q i ą p U 2 p n k |x 0 qu ,
where K hn p.q :" Kp.{h n q{h d n , with K a joint density function on R d , ph n q ně1 is a positive nonrandom sequence of bandwidths with h n Ñ 0 if n Ñ 8, 1l A the indicator function on the event A, and p U j p.|x 0 q is an estimator for U j p.|x 0 q, defined as p U j p.|x 0 q " infty : p F n,j py|x 0 q ě 1 ´1{.u, where p F n,j py|x 0 q is a classical kernel estimator [START_REF] Rosenblatt | Remarks on some nonparametric estimates of a density function[END_REF][START_REF] Parzen | On estimation of a probability density function and mode[END_REF], given by p F n,j py|x 0 q "

1 n

ř n i"1 K hn px 0 ´Xi q1l

tY pjq i ďyu p f n px 0 q , (3)
for j " 1, 2, with p f n px 0 q :" 1 n

n ÿ i"1
K hn px 0 ´Xi q, being a density estimator.

To study the asymptotic behaviour of this estimator we need to introduce some conditions.

We assume that Y p1q and Y p2q are positive random variables, and that they follow a conditional Pareto-type model. Let RV ψ denote the class of regularly varying functions at infinity with index ψ, i.e., positive measurable functions f satisfying f ptxq{f ptq Ñ x ψ , as t Ñ 8, for all x ą 0. If ψ " 0, then we call f a slowly varying function at infinity.

Assumption pDq For all x P S X , the conditional survival function of Y pjq , j " 1, 2, given X " x, satisfies F j py|xq " A j pxqy ´1{γ j pxq ˆ1 `1 γ j pxq δ j py|xq ˙,

where A j pxq ą 0, γ j pxq ą 0, and |δ j p.|xq| is normalized regularly varying at infinity with index ´βj pxq, β j pxq ą 0, i.e., δ j py|xq " B j pxq exp

ˆż y 1 ε j pu|xq u du ˙,
with B j pxq P R and ε j py|xq Ñ ´βj pxq as y Ñ 8. Moreover, we assume y Ñ ε j py|xq to be a continuous function.

Under Assumption pDq we have that U j p¨|xq, j " 1, 2, satisfy U j py|xq " rA j pxqs γ j pxq y γ j pxq p1 `aj py|xqq ,

where a j py|xq :" δ j pU j py|xq|xqp1 `op1qq, and thus |a j p.|xq| P RV ´βj pxqγ j pxq .

We also need a condition that describes the right-hand upper tail dependence of the joint distribution of pY p1q , Y p2q q given X " x. Let R t py 1 , y 2 |xq :" tPpF 1 pY p1q |xq ď y 1 {t, F 2 pY p2q |xq ď y 2 {t|X " xq.

Assumption pRq For all x P S X we have as t Ñ 8 R t py 1 , y 2 |xq Ñ Rpy 1 , y 2 |xq, uniformly in y 1 , y 2 P p0, T s, for any T ą 0, and in x P Bpx 0 , rq, for some r ą 0.

Note that this condition reflects in fact the asymptotic behavior of the conditional copula function.

It is an adjustment of the usual first order condition in multivariate extreme value theory (see, e.g., Chapter 6 in de Haan and[START_REF] De Haan | Extreme value theory, an introduction[END_REF][START_REF] Cai | Estimation of the marginal expected shortfall: the mean when a related variable is extreme[END_REF] to the regression context.

Based on conditions pDq and pRq we can already obtain the following theoretical approximation to θ β,p px 0 q, which is a result that is in fact valid for any p Ó 0.

Lemma 1 Assume pDq, pRq and γ 1 px 0 q ă 1{β. Then, as p Ó 0 we have θ β,p px 0 q rU 1 p1{p|x 0 qs β Ñ ´ż 8 0 R ˜"1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q , 1 ˇˇx 0 ¸du ´γ1 px 0 q .

In order to deal with the regression context, f X p.q, Rpy 1 , y 2 |.q and the functions appearing in F j py|.q, j " 1, 2, are assumed to satisfy the following Hölder conditions. Let }.} denote some norm on R d .

Assumption pHq There exist positive constants M f X , M R , M A j , M γ j , M B j , M ε j , η f X , η R , η A j , η γ j , η B j , η ε j , where j " 1, 2, and κ ą γ 1 px 0 qβ, such that for all x, z P S X :

|f X pxq ´fX pzq| ď M f X }x ´z} η f X , sup y 1 ą0, 1 2 ďy 2 ď2 |Rpy 1 , y 2 |xq ´Rpy 1 , y 2 |zq| y κ 1 ^1 ď M R }x ´z} η R , |A j pxq ´Aj pzq| ď M A j }x ´z} η A j , |γ j pxq ´γj pzq| ď M γ j }x ´z} ηγ j , |B j pxq ´Bj pzq| ď M B j }x ´z} η B j , sup yě1 |ε j py|xq ´εj py|zq| ď M ε j }x ´z} ηε j .
We also impose a condition on the kernel function K, which is a standard condition in local estimation, see, e.g., [START_REF] Daouia | Kernel estimators of extreme level curves[END_REF][START_REF] Daouia | On kernel smoothing for extremal quantile regression[END_REF] and Escobar-Bach et al. (2018a).

Assumption pKq K is a bounded density function on R d , with support S K included in the unit ball in R d , with respect to the norm }.}.

Before stating the weak convergence of p θ n , we need to introduce a second order condition.

Assumption pSq. There exist κ ą γ 1 px 0 qβ and τ ă 0 such that, as t Ñ 8

sup xPBpx 0 ,rq sup y 1 ą0, 1 2 ďy 2 ď2 |R t py 1 , y 2 |xq ´Rpy 1 , y 2 |xq| y κ 1 ^1 " Opt τ q,
for some r ą 0.

This second order condition specifies the rate of convergence of R t py 1 , y 2 |xq to its limit Rpy 1 , y 2 |xq as t Ñ 8. It is an adjustment of the second order condition in [START_REF] Cai | Estimation of the marginal expected shortfall: the mean when a related variable is extreme[END_REF] to the regression context, by assuming that the order of approximation is also uniform in x P Bpx 0 , rq. Note that the uniform requirement in the second order condition excludes the case where pY p1q , Y p2q q are asymptotically upper tail independent given X " x 0 , which corresponds to Rpy 1 , y 2 |x 0 q " 0.

We can now state our first main result, the weak convergence of p θ n { p f n px 0 q, properly normalized. Weak convergence is denoted by the arrow .

Theorem 1 Assume pDq, pHq, pKq, pSq with x Ñ Rpy 1 , y 2 |xq being a continuous function, Rpy, 1|x 0 q is continuously differentiable in y, and y Ñ F j py|x 0 q, j " 1, 2, are strictly increasing. Let x 0 P IntpS X q such that f X px 0 q ą 0. Consider sequences k " tn α 1 pnqu and h n " n ´∆ 2 pnq, where 1 and 2 are slowly varying functions at infinity, with α P p0, 1q and

max ˆα d `2pη R ^ηf X ^ηA 1 ^ηγ 1 q , α ´2β 1 px 0 qγ 1 px 0 qp1 ´αq d , α d `2pη A 2 ^ηγ 2 qr1 ´βγ 1 px 0 qs , α ´2β 2 px 0 qγ 2 px 0 qp1 ´αqp1 ´βγ 1 px 0 qq d `2pη ε 2 ^ηB 2 qr1 ´βγ 1 px 0 qs , α d ´2βp1 ´αqγ 2 1 px 0 qβ 1 px 0 q dr1 `γ1 px 0 qpβ 1 px 0 q ´εqs , α `2p1 ´αqτ q d ˙ă ∆ ă α d ,
where 0 ă ε ă β 1 px 0 q. Then, for γ 1 px 0 q ă 1{p2βq, we have

b kh d n ˜p θ n p f n px 0 qθ β,k{n px 0 q ´1¸ 1 f X px 0 q ş 8 0 Rpr1 `u´γ 1 px 0 q β s ´1 γ 1 px 0 q , 1|x 0 qdu ´γ1 px 0 q ˆ"ż 8 0 W pr1 `u´γ 1 px 0 q β s ´1 γ 1 px 0 q , 1qdu ´γ1 px 0 q
´pW p1, 8q ´W p8, 1qq

ż 8 0 R 1 pr1 `u´γ 1 px 0 q β s ´1 γ 1 px 0 q , 1|x 0 qp1 `u´γ 1 px 0 q β q ´1 γ 1 px 0 q du ´γ1 px 0 q * ´W p8, 1q f X px 0 q `βγ 1 px 0 q f X px 0 q W p1, 8q,
where W py 1 , y 2 q is a zero centered Gaussian process with covariance function

E pW py 1 , y 2 qW py 1 , y 2 qq " }K} 2 2 f X px 0 qRpy 1 ^y1 , y 2 ^y2 |x 0 q, with }K} 2 :" b ş R d K 2 puqdu
, and W p8, yq and W py, 8q are zero centered Gaussian processes with the same covariance function EpW p8, yqW p8, yqq " EpW py, 8qW py, 8qq " }K} 2 2 f X px 0 q py ^yq , and R 1 py, 1|x 0 q denotes the derivative of Rpy, 1|x 0 q with respect to y.

The variance of the limiting random variable in Theorem 1, denoted W 1 , is given by

V arpW 1 q " }K} 2 2 f X px 0 q " β 2 γ 2 1 px 0 q ´1 ´I1 I 2 2 `2 r1 ´Rp1, 1|x 0 qs ˆ˜ˆI 3 I 2 ˙2 ´r1 `βγ 1 px 0 qs I 3 I 2 `βγ 1 px 0 q ¸+ ,
where

I 1 :" ż 8 0 R ˆ"1 `u´γ 1 px 0 q{β ı ´1{γ 1 px 0 q , 1 ˇˇx 0 ˙du ´2γ 1 px 0 q , I 2 :" ż 8 0 R ˆ"1 `u´γ 1 px 0 q{β ı ´1{γ 1 px 0 q
, 1 ˇˇx 0 ˙du ´γ1 px 0 q , I 3 :"

ż 8 0 R 1 ˆ"1 `u´γ 1 px 0 q{β ı ´1{γ 1 px 0 q , 1 ˇˇx 0 ˙"1 `u´γ 1 px 0 q{β ı ´1{γ 1 px 0 q du ´γ1 px 0 q .
3 Estimator for the extreme case

We now turn to the estimation of θ β,p px 0 q under extrapolation. Assuming pDq, pRq and γ 1 px 0 q ă 1{β, according to Lemma 1, we have the following approximation

θ β,p px 0 q " rU 1 p1{p|x 0 qs β rU 1 pn{k|x 0 qs β θ β, k n px 0 q " ˆk np ˙βγ 1 px 0 q θ β, k n px 0 q.
To estimate θ β,p px 0 q, we can use the estimator θ β, k n px 0 q :" p θ n { p f n px 0 q of θ β, k n px 0 q combined with an estimator of the extreme value index γ 1 px 0 q. For the latter, we can use the local [START_REF] Hill | A simple general approach to inference about the tail of a distribution[END_REF] estimator

p γ 1,k 1 px 0 q :" 1 k 1 ř n i"1 K hn px 0 ´Xi q ´ln Y p1q i ´ln p U 1 pn{k 1 |x 0 q ¯1l tY p1q i ě p U 1 pn{k 1 |x 0 qu p f n px 0 q ,
already studied in [START_REF] Goegebeur | A Weissman-type estimator of the conditional marginal expected shortfall[END_REF], and based on an intermediate sequence k 1 , possibly different to k, such that k 1 Ñ 8 with k 1 {n Ñ 0. This yields the Weissman-type estimator for θ β,p px 0 q p θ β,p px 0 q " ˆk np

˙βp γ 1,k 1 px 0 q θ β, k n px 0 q.
The estimator is said to be of Weissman-type, as it is in nature similar to an estimator for an extreme quantile proposed by [START_REF] Weissman | Estimation of parameters and large quantiles based on the k largest observations[END_REF].

The next theorem states our main result, namely the weak convergence of our Weissman-type estimator p θ β,p px 0 q, properly normalized.

Theorem 2 Assume pDq, pHq, pKq, pSq with x Ñ Rpy 1 , y 2 |xq being a continuous function, Rpy, 1|x 0 q is continuously differentiable in y, and y Ñ F j py|x 0 q, j " 1, 2, are strictly increasing. Let x 0 P IntpS X q such that f X px 0 q ą 0. Consider sequences k " tn α 1 pnqu, k 1 " tn α 1 2 pnqu and h n " n ´∆ 3 pnq, where 1 , 2 and 3 are slowly varying functions at infinity, with α P p0, 1q and

α ď α 1 ă min ˆα d rd `2 pη f X ^ηA 1 ^ηγ 1 qs , α `2γ 1 px 0 qβ 1 px 0 q 1 `2γ 1 px 0 qβ 1 px 0 q ˙, and
max ˆα d `2pη R ^ηf X ^ηA 1 ^ηγ 1 q , α ´2β 1 px 0 qγ 1 px 0 qp1 ´αq d , α d `2pη A 2 ^ηγ 2 qr1 ´βγ 1 px 0 qs , α ´2β 2 px 0 qγ 2 px 0 qp1 ´αqp1 ´βγ 1 px 0 qq d `2pη ε 2 ^ηB 2 qr1 ´βγ 1 px 0 qs , α d ´2βp1 ´αqγ 2 1 px 0 qβ 1 px 0 q dr1 `γ1 px 0 qpβ 1 px 0 q ´εqs , α `2p1 ´αqτ q d , α 1 d `2pη f X ^ηγ 1 ^ηA 1 q , α 1 ´2γ 1 px 0 qβ 1 px 0 qp1 ´α1 q d ˙ă ∆ ă α d ,
where 0 ă ε ă β 1 px 0 q.

Then, for γ 1 px 0 q ă 1{p2βq and p satisfying p ď k n such that ln k{pnpq ?

k 1 h d n Ñ 0 and b k k 1 ln k np Ñ r P r0, 8s, we have min ˜bkh d n , a k 1 h d n ln k{pnpq ¸˜p θ β,p px 0 q θ β,p px 0 q ´1¸ minpr, 1q βγ 1 px 0 q f X px 0 q ˆż 1 0 W pu, 8q 1 u du ´W p1, 8q ṁin ˆ1, 1 r ˙$ & % 1 f X px 0 q ş 8 0 Rpr1 `u´γ 1 px 0 q β s ´1 γ 1 px 0 q , 1|x 0 qdu ´γ1 px 0 q ˆ"ż 8 0 W pr1 `u´γ 1 px 0 q β s ´1 γ 1 px 0 q , 1qdu ´γ1 px 0 q
´pW p1, 8q ´W p8, 1qq

ż 8 0 R 1 pr1 `u´γ 1 px 0 q β s ´1 γ 1 px 0 q , 1|x 0 qp1 `u´γ 1 px 0 q β q ´1 γ 1 px 0 q du ´γ1 px 0 q  ´W p8, 1q f X px 0 q `βγ 1 px 0 q f X px 0 q W p1, 8q
* .

The variance of the limiting random variable in Theorem 2, denoted W 2 , is given by

V arpW 2 q " }K} 2 2 f X px 0 q rminpr, 1qs 2 β 2 γ 2 1 px 0 q `"min ˆ1, 1 r ˙2 « β 2 γ 2 1 px 0 q ´1 ´I1 I 2 2 `2 r1 ´Rp1, 1|x 0 qs ˜ˆI 3 I 2 ˙2 ´r1 `βγ 1 px 0 qs I 3 I 2 `βγ 1 px 0 q ¸ff `2 min ˆr, 1 r ˙βγ 1 px 0 q « 1 β I 4 `I5 γ 1 px 0 qI 2 `ˆRp1, 1|x 0 q ´ż 1 0 Rpu, 1|x 0 q u du ˙ˆ1 ´I3 I 2 ˙´1 ff+ ,
where I 1 , I 2 and I 3 are defined as in the definition of the variance of W 1 and

I 4 :" ż 8 0 R ˆ"1 `u´γ 1 px 0 q{β ı ´1{γ 1 px 0 q , 1 ˇˇx 0 ˙u´γ 1 px 0 q{β 1 `u´γ 1 px 0 q{β du ´γ1 px 0 q , I 5 :" ż 8 0 R ˆ"1 `u´γ 1 px 0 q{β ı ´1{γ 1 px 0 q , 1 ˇˇx 0 ˙lnp1 `u´γ 1 px 0 q{β qdu ´γ1 px 0 q .

Simulation

In this section we illustrate the finite sample performance of p θ β,p px 0 q with a simulation experiment. We simulate from the following models: Model 1. We consider the logistic copula model Cpu 1 , u 2 |xq " e ´rp´ln u 1 q x `p´ln u 2 q x s 1{x , u 1 , u 2 P r0, 1s, x ě 2.

(

) 5
We take X " U r2, 10s, and combine this copula model with a Burrpζ, λ, τ q distribution for Y p1q : `yx 2 q 1{x , τ " ´1 and κ " 1 ´ε for some small ε ą 0.

F 1 pyq " 1 ´ˆζ ζ `yτ ˙λ , y ą 0; ζ, λ, τ ą 0, with ζ " 1, λ " 0.
Model 2. The conditional distribution of pY p1q , Y p2q q given X " x is that of

p|Z 1 | γ 1 pxq , |Z 2 | γ 2 pxq q,
where pZ 1 , Z 2 q follow a bivariate standard Cauchy distribution with density function

f pz 1 , z 2 q " 1 2π p1 `z2 1 `z2 2 q ´3{2 , pz 1 , z 2 q P R 2 .
We take X " U r0, 1s, and set γ 1 pxq " 0.25 `0.125 sinp2πxq, γ 2 pxq " 0. Model 3: We consider again the conditional logistic copula model defined in (5) combined with conditional Burr distributions for Y p1q and Y p2q given X " x, where we take ζ 1 " ζ 2 " 1, λ 1 " 1, λ 2 " 0.5, and τ 1 pxq " 2e 0.2x , τ 2 pxq " 8{ sinp0.3xq.

For this model γ 1 pxq " 0.5e ´0.2x . Similarly to Model 1, this model satisfies pSq.

For all the models, the conditional marginal distributions F j p.|x 0 q, j " 1, 2, satisfy Assumption pDq (see, e.g., [START_REF] Beirlant | Second-order refined peaks-over-threshold modelling for heavy-tailed distributions[END_REF], Table 1), as well as Assumption pHq.

We implement our estimators p θ β,p px 0 q and p γ 1,k 1 px 0 q with a bi-quadratic kernel function, given by Kpxq " 15 16 p1 ´x2 q 2 1l txPr´1,1su , which clearly satisfies Assumption pKq. Related to this, we need to select also a bandwidth h n .

To this aim, we use the cross-validation procedure introduced by Yao (1999), and already used in the extreme value framework by [START_REF] Daouia | Kernel estimators of extreme level curves[END_REF][START_REF] Daouia | On kernel smoothing for extremal quantile regression[END_REF] and Escobar-Bach et al. (2018a), and defined as:

h cv :" argmin hnPH n ÿ i"1 n ÿ j"1 ˆ1l ! Y p2q i ďY 2q j ) ´p F n,hn,2,´i ´Y p2q j ˇˇX i ¯˙2 ,
where H is the grid of values defined as R X ˆt0.05, 0.10, . . . , 0.30u, with R X the range of the covariate X, and p F n,hn,2,´i py|xq :"

ř n k"1,k‰i K hn px ´Xk q 1l ! Y p2q k ďy ) ř n k"1,k‰i K hn px ´Xk q .
We simulate 500 datasets of sizes n " 500 and 1 000 from each model. For each sample, we compute p θ β,p px 0 q for two different values of p: 1{n and 1{p2nq, and for two values of β, where the latter is chosen such that the condition γ 1 px 0 q ă 1{p2βq is satisfied: β " 1 for all models and β " 1.5, 1.25, 1.2, for Models 1-3, respectively.

The intermediate sequence k 1 on which the estimator for γ 1 px 0 q is based is selected by a graphical assessment, where k 1 is chosen as the smallest value after which the median of p γ 1,k 1 px 0 q, computed over the 500 replications, shows a stable part.

In Figure 1 we show for Model 1 the boxplots of p θ 1,p px 0 q, where k " 0.15 n, at various positions of the covariate x 0 , for n " 500 (top row) and n " 1 000 (bottom row), and for p " 1{n (left) and p " 1{p2nq (right). The red curve shows the true value of θ 1,p px 0 q. The layout of Figure 2 is similar but with β " 1.5. Figures 3 and4 show the corresponding results for Model 2, and Figures 5 and6 for Model 3. From these simulations we can draw the following conclusions:

• Overall, the estimator performs quite well, but of course the results depend on the model and the covariate position. In Model 1, Rpy 1 , y 2 |x 0 q depends on the covariate, but the marginal distributions do not. For Model 2, Rpy 1 , y 2 |x 0 q does not depend on the value of x 0 but the marginal distributions do, and for Model 3 both Rpy 1 , y 2 |x 0 q and the two marginals depend on x 0 . As is clear from the plots of the true function θ β,p px 0 q, this quantity does not change a lot in the covariate for Model 1, while on the contrary it changes a lot in the covariate for Model 2, where there is a maximum and a minimum. In Model 3, θ β,p px 0 q decreases in x 0 . Hence, the estimation under Model 1 is easier than for the other two models.

• The function θ β,p px 0 q follows the pattern of γ 1 px 0 q rather closely, see Figure 7 where we graph the functions γ 1 px 0 q for Models 2 and 3.

• Of the three models considered, Model 2 is most challenging for estimation as θ β,p px 0 q changes a lot with the covariate. Near the x 0 value where θ β,p px 0 q attains its maximum, the method tends to underestimate the true value. This can be explained by the local nature of the estimation: indeed, for such positions, local estimation will be based on Y p1q data coming from distributions with a lighter tail, leading to an underestimation.

• Note that the estimation with p " 1{n corresponds already with extrapolation, as the estimation is done locally, and hence based on fewer observations than n.

• For a given n, smaller values of p lead, as expected, to more variable estimates, as do larger values of β.

• The variability of p θ β,p px 0 q is larger at x 0 where γ 1 px 0 q is large. This can be expected since at such positions the Y p1q -data are more heavy tailed, and hence show large variability.

• The cross-validation procedure that was used here leads to a global bandwidth, which gives a reasonable performance for the whole covariate range, and which also works quite well for a wide variety of models. For situations where θ β,p px 0 q varies a lot with the covariate it can be advantageous to use a local bandwidth, since at such positions a smaller bandwidth will lead to less bias in the estimation. Evaluating the performance of bandwidth selection criteria is a topic for future research.

Real data analysis

In this section we illustrate our method on a real dataset. We analyze the dataset Vehicle Insurance Customer Data, which is publicly available at https://www.kaggle.com/ranja7/vehicle-insurance-customer-data. The dataset contains socio-economic data of insurance customers as well as details about the insured vehicle. We focus on estimating the first and second marginal excess moments of the cumulative claim amount throughout the contract, Y p1q , given a customer's lifetime value, Y p2q , exceeding a high quantile of its distribution and given an observation for the covariate income, X. In our analysis, we only consider the data for which the customer's income is not zero, which leads to a total of n " 6817 observations. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 3.5 4 4.5 5 5.5 6 6. CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 3.5 4 4.5 5 5.5 6 6. CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 3.5 4 4.5 5 5.5 6 6. CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 3.5 4 4.5 5 5.5 6 6. Boxplots of p θ 1,p px 0 q for n " 500 (top row) and n " 1 000 (bottom row), and p " 1{n (left) and p " 1{p2nq (right), along with θ 1,p px 0 q (red curve). q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 3.5 4 4.5 5 5.5 6 6. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 3.5 4 4.5 5 5.5 6 6. CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 3.5 4 4.5 5 5.5 6 6. CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 3.5 4 4.5 5 5.5 6 6. Boxplots of p θ 1.5,p px 0 q for n " 500 (top row) and n " 1 000 (bottom row), and p " 1{n (left) and p " 1{p2nq (right), along with θ 1.5,p px 0 q (red curve). q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 0 5 10 15 20 x0 CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 0 5 10 15 20 x0 CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 0 5 10 15 20 x0 CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.05 0. Boxplots of p θ 1,p px 0 q for n " 500 (top row) and n " 1 000 (bottom row), and p " 1{n (left) and p " 1{p2nq (right), along with θ 1,p px 0 q (red curve). q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 0 10 20 30 40 x0 CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 0 10 20 30 40 x0 CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 0 10 20 30 40 x0 CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.05 0. 25,p px 0 q for n " 500 (top row) and n " 1 000 (bottom row), and p " 1{n (left) and p " 1{p2nq (right), along with θ 1.25,p px 0 q (red curve). q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 3. CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 3. CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 3. CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 3. Boxplots of p θ 1,p px 0 q for n " 500 (top row) and n " 1 000 (bottom row), and p " 1{n (left) and p " 1{p2nq (right), along with θ 1,p px 0 q (red curve). q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 0 5 10 15 x0 CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 0 5 10 15 x0 CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 0 5 10 15 x0 CMEM q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 3. 2,p px 0 q for n " 500 (top row) and n " 1 (bottom row), and p " 1{n (left) and p " 1{p2nq (right), along with θ 1.2,p px 0 q (red curve). In Figure 8, we show a scatterplot of the total claim amount versus the customer's lifetime value, which suggests a positive dependence between the two variables. The model assumptions of asymptotic dependence and marginal conditional distributions of Pareto-type with moreover γ 1 px 0 q ă 1{p2βq can be verified by estimating Rp1, 1|x 0 q, the construction of local Pareto quantile plots for the Y p1q and Y p2q data, and the estimation of γ 1 px 0 q. We refer for this analysis to Goegebeur et al. (2021b), where in particular γ 1 px 0 q estimates were obtained in the range 0.30-0.35, and this for several values of x 0 .

From Theorem 2, by applying the log-transformation, we obtain the following pointwise p1 ´αqconfidence interval for θ 1,p px 0 q:

« p θ 1,p px 0 q exp # z α{2 a V arpW 2 q a n + , p θ 1,p px 0 q exp # ´zα{2 a V arpW 2 q a n +ff ,
where z α{2 is the α{2-quantile of the standard normal distribution and a n :" min ˆakh d n ,

?

k 1 h d n ln k{pnpq ˙.
Note that a confidence interval for θ 2,p px 0 q cannot be obtained due to the condition γ 1 px 0 q ă 1{4 in Theorem 2, which is not satisfied for the present data. However, it can be shown that for consistency of p θ β,p px 0 q one has the weaker constraint γ 1 px 0 q ă 1{β, meaning that for the present data we can still estimate θ 2,p px 0 q in a consistent way.

As for the confidence intervals, we estimate the integrals involved in the variance of the limiting random variable W 2 by replacing the function R with an adjusted version of the estimator proposed by Escobar-Bach et al. (2018b) for the stable tail dependence function: k ln k np . As for the choice of the bandwidth h n , we again determine it by applying the cross-validation criterion described in Section 4.

p Rpy 1 , y 2 |x 0 q " 1 k ř n i"1 K hn px 0 ´Xi q 1l ! p F n,1 pY p1q i |X i qď k n y 1 , p F n,2 pY p2q i |X i qď k n y 2 ) p f n px 0 q , where p F n,
In Figure 9, we show the estimates for θ 1,p px 0 q together with the pointwise 95%-confidence intervals for different values of income x 0 , with p " 0.001 and p " 0.0005. As expected, the estimates of θ 1,p px 0 q with p " 0.0005 are higher than those with p " 0.001, while at the same time p " 0.0005 leads also to wider confidence intervals than p " 0.001, reflecting the higher uncertainty of the estimate, which is due to the fact that the estimation is based on fewer observations. Combining these with the estimates for second marginal excess moments we obtain estimates for the standard deviation of the excess pY p1q ´U1 p1{p|x 0 qq `, given that Y p2q ą U 2 p1{p|x 0 q and X " x 0 , which is shown in Figure 10.

Preliminary results

In order to prove our main theorems, we need some auxiliary results. For any u ą 0 and v P S K , define

r s n py 1 , u|x 0 ´hn vq " n k F 1 ˆU1 ´n k ˇˇx 0 ¯y1 " 1 `u´γ 1 px 0 q β  ˇˇx 0 ´hn v ˙,
q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 20000 40000 60000 80000 Income CMEM q q q q q q q q q q q q q q q q q q q 0 500 1000 1500 2000 2500 3000 10000 25000 40000 55000 70000 85000 100000 Income CMEM q q q q q q q q q q q q q q q q q q q 0 500 1000 1500 2000 2500 3000 10000 25000 40000 55000 70000 85000 100000

Figure 9: Vehicle insurance dataset. Estimates for θ 1,p px 0 q together with the pointwise 95%confidence interval for p " 0.001 (left) and p " 0.0005 (right). Figure 10: Vehicle insurance dataset. Estimates for the standard deviation of pY p1q ´U1 p1{p|x 0 qq given that Y p2q ą U 2 p1{p|x 0 q and X " x 0 as function of income for p " 0.001 (solid line) and p " 0.0005 (dashed line). and

r t n py 2 |x 0 ´hn vq " n k F 2 ˆU2 ˆn ky 2 ˇˇx 0 ˙ˇˇx 0 ´hn v ˙.
Lemma 2 Assume pDq and pHq and x 0 P IntpS X q. Consider sequences k Ñ 8 and h n Ñ 0 as n Ñ 8, in such a way that k{n Ñ 0 and h ηε j ^ηγ j n ln n k Ñ 0, j " 1, 2. Then, we have, for n large, that ˇˇˇˇr s n py 1 , u|x 0 ´hn vq ´ˆy 1

" 1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q ˇˇˇď C " 1 `u´γ 1 px 0 q β  ´1{γ 1 px 0 q # h η A 1 n `hηγ 1 n ln n k `hηγ 1 n " 1 `u´γ 1 px 0 q β  Ch ηγ 1 n ln ˆ1 `u´γ 1 px 0 q β δ1 ´U1 ´n k ˇˇx 0 ¯ˇˇx 0 ¯ˇˇˆ« 1 `hηε 1 n " 1 `u´γ 1 px 0 q β  Ch ηε 1 n ˆln n k `ln ˆ1 `u´γ 1 px 0 q β ˙ḣ ηε 1 `ηγ 1 n " 1 `u´γ 1 px 0 q β  Ch ηε 1 n `Ch ηγ 1 n ln ˆ1 `u´γ 1 px 0 q β ˙ˆln n k `ln ˆ1 `u´γ 1 px 0 q β ˙˙ff+ and ˇˇr t n py 2 |x 0 ´hn vq ´y2 ˇˇď C ! h η A 2 n `hηγ 2 n ln n k `ˇˇδ 2 ´U2 ´n k ˇˇx 0 ¯ˇˇx 0 ¯ˇˇ" h η B 2 n `hηε 2 n ln n k ı) ,
uniformly in u ą 0, y 1 , y 2 P r1{2, 2s and in v P S K .

Lemma 3 Assume pDq, pHq, γ 1 px 0 q ă 1{β and x 0 P IntpS X q. For sequences k " tn α 1 pnqu and h n " n ´∆ 2 pnq, where 1 and 2 are slowly varying functions at infinity, with α P p0, 1q and , y 2 ˇˇx 0 ¸ff du ´γ1 px 0 q ˇˇˇˇÑ 0.

max ˆα d `2pη A 1 ^ηγ 1 q , α ´2β
Introduce now the notation p θ n py 1 , y 2 q :" 1 k

n ÿ i"1 K hn px 0 ´Xi q " Y p1q i ´U1 ´n k |x 0 ¯y1 ı β `1l tY p2q i ąU 2 p n ky 2 |x 0 qu .
Assuming F 1 py|x 0 q strictly increasing in y, we have

p θ n py 1 , y 2 q " ´"y 1 U 1 ´n k |x 0 ¯ıβ ż 8 0 T n pp s n py 1 , u|x 0 q, y 2 |x 0 q du ´γ1 px 0 q , (6)
where

T n py 1 , y 2 |x 0 q :" 1 k n ÿ i"1 K hn px 0 ´Xi q1l tF 1 pY p1q i |x 0 qď k n y 1 ,F 2 pY p2q i |x 0 qď k n y 2 u and p s n py 1 , u|x 0 q :" n k F 1 ˆU1 ´n k |x 0 ¯y1 " 1 `u´γ 1 px 0 q β  ˇˇx 0 ˙.
Remark that our final estimator p θ n " p θ n pp u n,1 , p e n,2 q , where p u n,i :" p U i pn{k|x 0 q U i pn{k|x 0 q , i " 1, 2, and p e n,2 :"

n k F 2 pp u n,2 U 2 pn{k|x 0 q|x 0 q .
Define now E n py 1 , y 2 q " E ˜p θ n py 1 , y 2 q ry 1 U 1 `n k |x 0 ˘sβ ¸.

Our next proposition establishes the uniform asymptotic behaviour of p θ n py 1 , y 2 q in y 1 , y 2 P r1{2, 2s, after proper normalization. The proof of the proposition uses a process convergence result for T n py 1 , y 2 |x 0 q from Goegebeur et al. (2021a), which is repeated below for completeness.

Theorem 3 [START_REF] Goegebeur | Conditional marginal expected shortfall[END_REF] Assume pDq, pHq, pKq, pRq with x Ñ Rpy 1 , y 2 |xq being a continuous function, x 0 P Int(S X q with f X px 0 q ą 0, and y Ñ F j py|x 0 q, j " 1, 2, are strictly increasing. Consider sequences k Ñ 8 and h n Ñ 0 as n Ñ 8, in such a way that k{n Ñ 0, kh d n Ñ 8 and h

ηγ 1 ^ηγ 2 ^ηε 1 ^ηε 2 n
ln n{k Ñ 0. Then for η P r0, 1{2q, we have,

b kh d n T n py 1 , y 2 |x 0 q ´EpT n py 1 , y 2 |x 0 qq y η 1 W py 1 , y 2 q y η 1 ,
in Dpp0, T s 2 q, for any T ą 0.

For convenient representation, the limiting process in Theorem 3 will be defined on the same probability space as T n py 1 , y 2 |x 0 q, via the Skorohod construction, but it should be kept in mind that it is only in distribution equal to the original process. The Skorohod representation theorem gives then (with keeping the same notation) sup y 1 ,y 2 Pp0,T s ˇˇˇˇa kh d n rT n py 1 , y 2 |x 0 q ´EpT n py 1 , y 2 |x 0 qqs ´W py 1 , y 2 q y η 1 ˇˇˇˇÑ 0, a.s. . Proposition 1 Assume pDq, pHq, pKq, pRq with x Ñ Rpy 1 , y 2 |xq being a continuous function, x 0 P Int(S X q with f X px 0 q ą 0, and y Ñ F j py|x 0 q, j " 1, 2, are strictly increasing. Consider sequences k Ñ 8 and h n Ñ 0 as n Ñ 8, in such a way that k{n Ñ 0, kh d n Ñ 8 and h ηγ 1 ^ηγ 2 ^ηε 1 ^ηε 2 n ln n{k Ñ 0. Then, for γ 1 px 0 q ă 1{p2βq, we have sup y1,y2Pr 1 2 ,2s ˇˇˇˇa kh d n rU 1 pn{k|x 0 qy 1 s β " p θ n py 1 , y 2 q ´E ´p θ n py 1 , y 2 q ¯ı `ż 8 0 W ˆ"y 1 ´1 `u´γ 1 px 0 q β ¯ı´1 γ 1 px 0 q , y 2 ˙du ´γ1px0q ˇˇˇP Ñ 0.

Proof of Lemma 2

According to Assumption pDq, using the inequality |e x ´1| ď |x| e |x| , we have for some positive constant C ˇˇˇˇr s n py 1 , u|x 0 ´hn vq ´ˆy 1

" 1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q ˇˇˇď ˇˇˇˇr s n py 1 , u|x 0 ´hn vq ´ˆy 1 " 1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 ´hnvq ˇˇˇČ " 1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q h ηγ 1 n " 1 `u´γ 1 px 0 q β  Ch ηγ 1 n ln ˆ1 `u´γ 1 px 0 q β ď C " 1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q ˆ#h ηγ 1 n " 1 `u´γ 1 px 0 q β  Ch ηγ 1 n ln ˆ1 `u´γ 1 px 0 q β 1 `hηγ 1 n " 1 `u´γ 1 px 0 q β  Ch ηγ 1 n ln ˆ1 `u´γ 1 px 0 q β ˙" ˇˇˇA 1 px 0 ´hn vq A 1 px 0 q ´1ˇˇˇˇ" U 1 ´n k ˇˇx 0 ¯ı 1 γ 1 px 0 q ´1 γ 1 px 0 ´hnvq ˆˇˇˇˇˇˇˇ1 `1 γ 1 px 0 ´hnvq δ 1 ˆU1 ´n k ˇˇx 0 ¯y1 " 1 `u´γ 1 px 0 q β  ˇˇx 0 ´hn v 1 `1 γ 1 px 0 q δ 1 ´U1 ´n k ˇˇx 0 ¯ˇˇx 0 ¯ˇˇˇˇˇˇ" U 1 ´n k ˇˇx 0 ¯ı 1 γ 1 px 0 q ´1 γ 1 px 0 ´hnvq ´1ˇˇˇˇˆˇˇˇˇˇˇˇ1 `1 γ 1 px 0 ´hnvq δ 1 ˆU1 ´n k ˇˇx 0 ¯y1 " 1 `u´γ 1 px 0 q β  ˇˇx 0 ´hn v 1 `1 γ 1 px 0 q δ 1 ´U1 ´n k ˇˇx 0 ¯ˇˇx 0 ¯ˇˇˇˇˇˇ1 `1 γ 1 px 0 ´hnvq δ 1 ˆU1 ´n k ˇˇx 0 ¯y1 " 1 `u´γ 1 px 0 q β  ˇˇx 0 ´hn v 1 `1 γ 1 px 0 q δ 1 ´U1 ´n k ˇˇx 0 ¯ˇˇx 0 ¯´1 ˇˇˇˇˇˇˇfi ffi ffi fl , / / . / / - .
Using Assumption pHq and again the inequality |e x ´1| ď |x| e |x| , we deduce that, for n large,

ˇˇˇA 1 px 0 ´hn vq A 1 px 0 q ´1ˇˇˇˇď C h η A 1 n (7) ˇˇˇ´U 1 ´n k ˇˇx 0 ¯¯1 γ 1 px 0 q ´1 γ 1 px 0 ´hnvq ´1ˇˇˇˇď C h ηγ 1 n ln n k . (8)
Now, using the explicit expression for δ 1 py|xq, Assumption pHq, and Proposition B.1.10 in de Haan and Ferreira (2006) with 0 ă ε ă β 1 px 0 q, we obtain, for n large, that

ˇˇˇˇˇˇˇ1 `1 γ 1 px 0 ´hnvq δ 1 ˆU1 ´n k ˇˇx 0 ¯y1 " 1 `u´γ 1 px 0 q β  ˇˇx 0 ´hn v 1 `1 γ 1 px 0 q δ 1 ´U1 ´n k ˇˇx 0 ¯ˇˇx 0 ¯´1 ˇˇˇˇˇˇď C ˇˇδ 1 ´U1 ´n k ˇˇx 0 ¯ˇˇx 0 ¯ˇˇ$ ' ' & ' ' % ˇˇˇγ 1 px 0 q γ 1 px 0 ´hn vq ´1ˇˇˇˇˇˇˇˇˇˇˇδ 1 ˆU1 ´n k ˇˇx 0 ¯y1 " 1 `u´γ 1 px 0 q β  ˇˇx 0 ´hn v δ1 ´U1 ´n k ˇˇx 0 ¯ˇˇx 0 ¯ˇˇˇˇˇˇδ 1 ˆU1 ´n k ˇˇx 0 ¯y1 " 1 `u´γ 1 px 0 q β  ˇˇx 0 ´hn v δ1 ˆU1 ´n k ˇˇx 0 ¯y1 " 1 `u´γ 1 px 0 q β  ˇˇx 0 ˙´1 ˇˇˇˇˇˇˇˇˇˇˇˇˇˇδ 1 ˆU1 ´n k ˇˇx 0 ¯y1 " 1 `u´γ 1 px 0 q β  ˇˇx 0 δ1 ´U1 ´n k ˇˇx 0 ¯ˇˇx 0 ¯ˇˇˇˇˇˇδ 1 ˆU1 ´n k ˇˇx 0 ¯y1 " 1 `u´γ 1 px 0 q β  ˇˇx 0 δ1 ´U1 ´n k ˇˇx 0 ¯ˇˇx 0 ¯´ˆy 1 " 1 `u´γ 1 px 0 q β ˙´β 1 px 0 q ˇˇˇˇˇˇy 1 " 1 `u´γ 1 px 0 q β ˙´β 1 px 0 q ´1ˇˇˇˇˇ+ ď C ˇˇδ 1 ´U1 ´n k ˇˇx 0 ¯ˇˇx 0 ¯ˇˇˆ« 1 `hηε 1 n " 1 `u´γ 1 px 0 q β  Ch ηε 1 n ˆln n k `ln ˆ1 `u´γ 1 px 0 q β ˙˙ff , (9)
uniformly in y 1 P r1{2, 2s and v P S K . Combining [START_REF] Daouia | On kernel smoothing for extremal quantile regression[END_REF], [START_REF] Daouia | Kernel estimators of extreme level curves[END_REF] with [START_REF] Das | Risk contagion under regular variation and asymptotic tail independence[END_REF], the first statement of Lemma 2 follows.

The proof of the second statement of Lemma 2 is similar, although simpler.

Proof of Lemma 3

We have the decomposition , y 2 ˇˇx 0 ¸ff du ´γ1 px 0 q ˇˇˇb kh d n ˇˇˇż

8 δn " R `r s n py 1 , u|x 0 ´hn vq, r t n py 2 |x 0 ´hn vq|x 0 Ȓ ˜ˆy 1 " 1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q
, y 2 ˇˇx 0 ¸ff du ´γ1 px 0 q ˇˇˇď b kh d n ˇˇˇż δn 0 R `r s n py 1 , u|x 0 ´hn vq, r t n py 2 |x 0 ´hn vq|x 0 ˘du ´γ1 px 0 q ˇˇb

kh d n ˇˇˇˇż δn 0 R ˜ˆy 1 " 1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q
, y 2 ˇˇx 0 ¸du ´γ1 px 0 q ˇˇˇb kh d n ż 8 0 ˇˇˇˇr s n py 1 , u|x 0 ´hn vq ´ˆy 1 " 1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q ˇˇˇˇd u ´γ1 px 0 q ´bkh d n ż 8 δn ˇˇr t n py 2 |x 0 ´hn vq ´y2 ˇˇdu ´γ1 px 0 q ": T 1 `T2 `T3 `T4 , where δ n is a sequence such that δ n Ñ 0. As for T 1 , since Rpy 1 , y 2 |x 0 q ď y 1 ^y2 , we have

T 1 ď ´bkh d n ż δn 0 r s n py 1 , u|x 0 ´hn vqdu ´γ1 px 0 q ď ´2 1 γ 1 px 0 q b kh d n ż δn 0 ˆ1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q du ´γ1 px 0 q ´bkh d n ż δn 0 ˇˇˇˇr s n py 1 , u|x 0 ´hn vq ´ˆy 1 " 1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q ˇˇˇˇd u ´γ1 px 0 q ": T 1,1 `T1,2 .
Clearly,

T 1,1 ď C b kh d n δ 1 β ´γ1 px 0 q n
, and, by using the first uniform bound from Lemma 2, and assuming that h

ηε 1 ^ηγ 1 n
| ln δ n | Ñ 0, we have, after tedious computations,

T 1,2 " o ˆbkh d n δ 1 β ´γ1 px 0 q n ˙,
leading to, for n large,

T 1 ď C b kh d n δ 1 β ´γ1 px 0 q n .
The term T 2 is similar, but simpler, and hence

T 2 ď C b kh d n δ 1 β ´γ1 px 0 q n .
Concerning T 3 , we can use again the first bound from Lemma 2, and obtain, for n large

T 3 ď C b kh d n ´hη A 1 n `hηγ 1 n ln n k `ˇˇδ 1 ´U1 ´n k ˇˇx 0 ¯ˇˇx 0 ¯ˇˇ¯,
while for T 4 we use the second bound from Lemma 2, and obtain, for n large

T 4 ď C b kh d n δ ´γ1 px 0 q n " h η A 2 n `hηγ 2 n ln n k `ˇˇδ 2 ´U2 ´n k ˇˇx 0 ¯ˇˇx 0 ¯ˇˇ´hηB 2 n `hηε 2 n ln n k ¯ı .
Take δ n " h ξ n . Remember k " tn α 1 pnqu and h n " n ´∆ 2 pnq, where 1 and 2 are slowly varying functions at infinity. Note that α P p0, 1q by the conditions k Ñ 8 and k{n Ñ 0. The upper bounds for T 1 , T 2 and T 4 , which must tend to zero, lead then to the following bounds for ξ ξ ă min ˆ2∆pη A 2 ^ηγ 2 q ´pα ´∆dq 2∆γ 1 px 0 q , 2β 2 px 0 qγ 2 px 0 qp1 ´αq `2∆pη ε 2 ^ηB 2 q ´pα ´∆dq 2∆γ 1 px 0 q ˙, and ξ ą α ´∆d 2∆r1{β ´γ1 px 0 qs .

For the parameter ∆, we obtain bounds from the above bounds for ξ, by requiring that the lower bound for ξ is always below the upper bound. This, combined with the bounds coming from the upper bound for T 3 and the condition kh d n Ñ 8, leads to

max ˆα d `2pη A 1 ^ηγ 1 q , α ´2β 1 px 0 qγ 1 px 0 qp1 ´αq d , α d `2pη A 2 ^ηγ 2 qr1 ´βγ 1 px 0 qs , α ´2β 2 px 0 qγ 2 px 0 qp1 ´αqp1 ´βγ 1 px 0 qq d `2pη ε 2 ^ηB 2 qr1 ´βγ 1 px 0 qs ˙ă ∆ ă α d .
So one can first choose α P p0, 1q, followed by selecting ∆ satisfying its bounds, and finally the ξ satisfying its bounds.

Proof of Proposition 1

Note that, using (6), we have

ˇˇˇˇa kh d n rU 1 pn{k|x 0 qy 1 s β " p θ n py 1 , y 2 q ´E ´p θ n py 1 , y 2 q ¯ı `ż 8 0 W ˜"y 1 ˆ1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q
, y 2 ¸du ´γ1 px 0 q ˇˇˇď ˇˇˇż 8 0 " b kh d n pT n pp s n py 1 , u|x 0 q, y 2 |x 0 q ´E pT n pp s n py 1 , u|x 0 q, y 2 |x 0 qqq ´W pp s n py 1 , u|x 0 q, y 2 qs du ´γ1 px 0 q ˇż 8 0 « W pp s n py 1 , u|x 0 q, y 2 q ´W ˜"y 1 ˆ1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q , y 2 ¸ff du ´γ1 px 0 q ˇˇˇ" : T 5 `T6 .

From Assumption pDq, one obtains for n ą n 0 that p s n py 1 , u|x 0 q ď 2 1`1{γ 1 px 0 q . Hence, for n ą n 0 and η P pγ 1 px 0 qβ, 1{2q T 5 ď sup y 1 Pp0,2

ˆ1

`u´γ 1 px 0 q β ˙´η γ 1 px 0 q u ´γ1 px 0 q´1 du.

The latter integral is finite since γ 1 px 0 q ă 1{p2βq. Using Theorem

ˇˇˇˇW ˜"y 1 ˆ1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q , y 2 ¸ˇˇˇ" y 1 ˆ1 `u´γ 1 px 0 q β ˙´η γ 1 px 0 q ˆsup y 1 Pr1{2,2s ˜´ż δ 0 " y 1 ˆ1 `u´γ 1 px 0 q β ˙´η γ 1 px 0 q du ´γ1 px 0 q şup y 1 ,y 2 Pr1{2,2s ˜´ż 8 δ
ˇˇˇˇW pp s n py 1 , u|x 0 q, y 2 q ´W ˜"y 1 ˆ1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q , y 2 ¸ˇˇˇˇd u ´γ1 px 0 q ¸.

Note that sup y 1 ,y 2 Pr1{2,2s,uďδ |W pp s n py 1 , u|x 0 q, y 2 q| rp s n py 1 , u|x 0 qs η ď sup y 1 Pp0,2 1`1{γ 1 px 0 q s,y 2 Pr1{2,2s

|W py 1 , y 2 q| y η 1 , for n ą n 0 , and sup ˆ´ż δ 0 rp s n py 1 , u|x 0 qs η du ´γ1 px 0 q ˙ď Cδ η{β´γ 1 px 0 q for n ą n 1 " Cε, by taking δ " ε 1{pη{β´γ 1 px 0 qq , and sup

y 1 ,y 2 Pr1{2,2s,uďδ ˇˇˇˇW ˜"y 1 ˆ1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q , y 2 ¸ˇˇˇ" y 1 ˆ1 `u´γ 1 px 0 q β ˙´η γ 1 px 0 q ď sup y 1 Pp0,2
y 1 Pr1{2,2s ˜´ż δ 0 " y 1 ˆ1
`u´γ 1 px 0 q β ˙´η γ 1 px 0 q du ´γ1 px 0 q ¸ď Cε. ˇˇˇˇW pp s n py 1 , u|x 0 q, y 2 q ´W ˜"y 1 ˆ1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q , y 2 ¸ˇˇˇˇd u ´γ1 px 0 q ḑ sup y 1 ,y 2 Pr1{2,2s,uěδ ˇˇˇˇW pp s n py 1 , u|x 0 q, y 2 q ´W ˜"y 1 ˆ1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q , y 2 ¸ˇˇˇˇδ ´γ1 px 0 q .

Since sup y 1 Pr1{2,2s,uą0 ˇˇˇˇp s n py 1 , u|x 0 q ´"y 1 ˆ1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q ˇˇˇˇ" op1q, and by using the uniform continuity of W on compact sets, one has for n ą n 2 that sup y 1 ,y 2 Pr1{2,2s,uěδ ˇˇˇˇW pp s n py 1 , u|x 0 q, y 2 q ´W ˜"y 1 ˆ1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q , y 2 ¸ˇˇˇˇď ε.

Combining the above one obtains that T 6 " o P p1q. This achieves the proof of Proposition 1.

7 Proof of the main results

Proof of Lemma 1

We have θ β,p px 0 q " β ż 8

U 1 p1{p|x 0 q " y 1 ´U1 ˆ1 p ˇˇx 0 ˙β´1 1 p P ˆY p1q ą y 1 , Y p2q ą U 2 ˆ1 p ˇˇx 0 ˙ˇˇX " x 0 ˙dy 1 " β " U 1 ˆ1 p ˇˇx 0 ˙β ż 8 1 rz ´1s β´1 1 p P ˆY p1q ą zU 1 ˆ1 p ˇˇx 0 ˙, Y p2q ą U 2 ˆ1 p ˇˇx 0 ˙ˇˇX " x 0 ˙dz. Note that lim pÓ0 1 p P ˆY p1q ą zU 1 ˆ1 p ˇˇx 0 ˙, Y p2q ą U 2 ˆ1 p ˇˇx 0 ˙ˇˇX " x 0 ˙" R ´z´1{γ 1 px 0 q , 1 ˇˇx 0 ¯,
thus we only need to interchange the limit with the integral. To this aim, we can apply the dominated convergence theorem since for z ě 1

1 p P ˆY p1q ą zU 1 ˆ1 p ˇˇx 0 ˙, Y p2q ą U 2 ˆ1 p ˇˇx 0 ˙ˇˇX " x 0 ˙ď min ˆ1, 1 p F 1 ˆzU 1 ˆ1 p ˇˇx 0 ˙ˇˇx 0 ˙" 1 p F 1 ˆzU 1 ˆ1 p ˇˇx 0 ˙ˇˇx 0 ˙,
and according to Proposition B.1.9.5 in de Haan and Ferreira (2006), for z ě 1, ε, δ ą 0, there exists a p 0 such that for p ă p 0 we have

1 p F 1 ˆzU 1 ˆ1 p ˇˇx 0 ˙ˇˇx 0 ˙ď p1 `δqz ´1 γ 1 px 0 q `ε.
This implies that, for p ă p 0 , θ β,p px 0 q rU 1 p1{p|x 0 qs β ď βp1 `δq ż 8 1 rz ´1s β´1 z ´1 γ 1 px 0 q `εdz " βp1 `δq ż 1 0 z β´1 p1 ´zq 1 γ 1 px 0 q ´β´ε´1 dz ă 8, as soon as ε P p0, 1{γ 1 px 0 q ´βq. This yields θ β,p px 0 q rU 1 p1{p|x 0 qs β Ñ β ż 8 1 rz ´1s β´1 R ´z´1{γ 1 px 0 q , 1 ˇˇx 0 ¯dz, from which Lemma 1 follows, after a change of variable.

Proof of Theorem 1

Our quantity of interest is

b kh d n ˜p θ n p f n px 0 qθ β,k{n px 0 q ´1¸" b kh d n ˜p θ n f X px 0 qθ β,k{n px 0 q ´1¸f X px 0 q p f n px 0 q ´1 p f n px 0 q c k n b nh d n ´p f n px 0 q ´fX px 0 q " b kh d n ˜p θ n f X px 0 qθ β,k{n px 0 q ´1¸p 1 `oP p1qq `oP p1q.
We have the decomposition

b kh d n ˜p θ n f X px 0 qθ β,k{n px 0 q ´1" b kh d n ´p u β n,1 ´1r
p u n,1 U 1 `n k |x 0 ˘sβ θ β,k{n px 0 q # a kh d n f X px 0 q ˜p θ n pp u n,1 , p e n,2 q rp u n,1 U 1 `n k |x 0 ˘sβ ´En pp u n,1 , p e n,2 q akh d n f X px 0 q pE n pp u n,1 , p e n,2 q ´En p1, 1qq `akh d n f X px 0 q ˜En p1, 1q ´fX px 0 qθ β,k{n px 0 q rU 1 `n k |x 0 ˘sβ ¸+ ": b kh d n ´p u β n,1 ´1¯`r p u n,1 U 1 `n k |x 0 ˘sβ θ β,k{n px 0 q tT 1,n `T2,n `T3,n u . (10)
This decomposition can be motivated by Proposition 1 and the convergences ´1¯ βγ 1 px 0 q f X px 0 q W p1, 8q, and from our Lemma 1, we have

b kh d n pp u n,1 ´1q
γ 1 px 0 q f X px 0 q W p1, 8q, b kh d n pp e n,2 ´1q ´W p8, 1q f X px 0 q , see Lemma
rp u n,1 U 1 `n k |x 0 ˘sβ θ β,k{n px 0 q P ÝÑ ´1 ş 8 0 Rpr1 `u´γ 1 px 0 q β s ´1 γ 1 px 0 q , 1|x 0 qdu ´γ1 px 0 q
.

We need to study the three terms T 1,n , T

rp u n,1 U 1 `n k |x 0 ˘sβ θ β,k{n px 0 q T 1,n 1 f X px 0 q ş 8 0 W pr1 `u´γ 1 px 0 q β s ´1 γ 1 px 0 q , 1qdu ´γ1 px 0 q ş 8 0 Rpr1 `u´γ 1 px 0 q β s ´1 γ 1 px 0 q , 1|x 0 qdu ´γ1 px 0 q . ( 11
)
Concerning T 2,n , according to (6)

T 2,n " a kh d n f X px 0 q "ż 8 0
E rT n pp s n p1, u|x 0 q, 1|x 0 qs du ´γ1 px 0 q ´ż 8 0 E rT n pp s n pp u n,1 , u|x 0 q, p e n,2 |x 0 qs du ´γ1 px 0 q * .

For y 1 either 1 or p u n,1 and y 2 either 1 or p e n,2 , we have

ż 8 0 E rT n pp s n py 1 , u|x 0 q, y 2 |x 0 qs du ´γ1 px 0 q " ż 8 0 R ˜y´1 γ 1 px 0 q 1 " 1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q , y 2 ˇˇx 0 ¸du ´γ1 px 0 q ż S K Kpvqf X px 0 ´hn vqdv `żS K Kpvq ż 8 0 " R `r s n py 1 , u|x 0 ´hn vq, r t n py 2 |x 0 ´hn vq|x 0 Ȓ ˜y´1 γ 1 px 0 q 1 " 1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q , y 2 ˇˇx 0 ¸ff du ´γ1 px 0 q f X px 0 ´hn vqdv `ż 8 0 ż S K Kpvq " R n k `r s n py 1 , u|x 0 ´hn vq,
´hn vq|x 0 ˘‰ f X px 0 ´hn vqdvdu ´γ1 px 0 q " f X px 0 q ż 8 0 R ˜y´1 γ 1 px 0 q 1 " 1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q , y 2 ˇˇx 0 ¸du ´γ1 px 0 q `OP ´hη f X n ¯`O P ´´n k ¯τ ¯`O P ph η R n q `oP ˜1 a kh d n ¸, (12)
by Assumptions pSq and pHq combined with Lemma 3.

This implies that where Hpzq :"

T 2,n " b kh d n "ż 8 0 R ˆ"1 `u´γ 1 px 0 q β ı ´1 γ 1 px 0 q , 1 ˇˇx0 ˙du ´γ1px0q
ż 8 0 R
˜z " 1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q , 1 ˇˇx 0 ¸du ´γ1 px 0 q .

Assuming Rpy, 1|x 0 q differentiable in y with a derivative R 1 py, 1|x 0 q, we have

H 1 pzq " ż 8 0 R 1
˜z " 1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q , 1 ˇˇx 0 ¸"1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q du ´γ1 px 0 q , by using the Lipschitz property of the R´function combined with the dominated convergence theorem. Then by the mean value theorem . Using again the dominated convergence theorem, we have that H 1 pzq is continuous at z " 1, by the assumed continuity of R 1 py, 1|x 0 q and the fact that R 1 py, 1|x 0 q is decreasing in y, Rpy, 1|x 0 q being concave. Then, by the continuous mapping theorem H 1 p p ξ n q P Ñ H 1 p1q. All in all, we have T 2,n W p8, 1q f X px 0 q ż 8 0 R ˜"1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q , 1 ˇˇx 0 ¸du ´γ1 px 0 q `W p1, 8q ´W p8, 1q f X px 0 q ż 8 0 R 1 ˜"1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q , 1 ˇˇx 0 ¸"1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q du ´γ1 px 0 q , from which we deduce that rp u n,1 U 1 `n k |x 0 ˘sβ θ β,k{n px 0 q T 2,n ´W p8, 1q f X px 0 q ´W p1, 8q ´W p8, 1q f X px 0 q ˆş8 0 R 1 pr1 `u´γ 1 px 0 q β s ´1 γ 1 px 0 q , 1|x 0 qr1 `u´γ 1 px 0 q β s ´1 γ 1 px 0 q du ´γ1 px 0 q ş 8 0 Rpr1 `u´γ 1 px 0 q β s ´1 γ 1 px 0 q , 1|x 0 qdu ´γ1 px 0 q .

Now, concerning T 3,n , we have T 3,n " a kh d n f X px 0 q " ´ż 8 0 E rT n pp s n p1, u|x 0 q, 1|x 0 qs du ´γ1 px 0 q ´fX px 0 qθ β,k{n px 0 q rU 1 pn{k|x 0 qs β * " b kh d n # ´ż 8 0 R ˜"1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q , 1 ˇˇx 0 ¸du ´γ1 px 0 q `ż 8 0 R n{k pp s n p1, u|x 0 q, 1|x 0 q du ´γ1 px 0 q * `op1q " b kh d n ż 8 0 " R n{k pp s n p1, u|x 0 q, 1|x 0 q ´R pp s n p1, u|x 0 q, 1|x 0 q ‰ du ´γ1 px 0 q `bkh d n ż 8 0 « R pp s n p1, u|x 0 q, 1|x 0 q ´R ˜"1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q , 1 ˇˇx 0 ¸ff du ´γ1 px 0 q `op1q ": T 3,1,n `T3,2,n `op1q.

As for T 3,1,n we use Assumption pSq to obtain R pp s n p1, u|x 0 q, 1|x 0 q ´R ˜"1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q , 1 ˇˇx 0 ¸ff du ´γ1 px 0 q ˇˇˇb kh d n ˇˇˇˇż 8 Tn « R pp s n p1, u|x 0 q, 1|x 0 q ´R ˜"1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q , 1 ˇˇx 0 ¸ff du ´γ1 px 0 q ˇˇˇˇ, where T n is a sequence that tends to infinity. Consequently, by using the Lipschitz continuity of Rp., .|x 0 q in the first integral and a straightforward bound for the second integral we have u ´γ1 px 0 q´1 ˇˇˇˇp s n p1, u|x 0 q ´"1 `u´γ 1 px 0 q β  ´1 γ 1 px 0 q ˇˇˇˇd u `2 b kh d n sup

y 1 ą0
Rpy 1 , 1|x 0 q T ´γ1 px 0 q n " op1q,

We have then rp u n,1 U 1 `n k |x 0 ˘sβ θ β,k{n px 0 q T 3,n " o P p1q.

To conclude the proof we comment on the sequence T n . Take T n " n ψ . Then, p14q is satisfied if ψ ą α ´∆d 2γ 1 px 0 q , and p15q is satisfied if ψ ă βr2p1 ´αqγ 1 px 0 qβ 1 px 0 q ´pα ´∆dqs 2r1 ´γ1 px 0 qpβ ´β1 px 0 q `εqs .

Requiring that the lower bound for ψ is below its upper bound gives the following constraint on ∆ ∆ ą α d ´2βp1 ´αqγ 2 1 px 0 qβ 1 px 0 q dr1 `γ1 px 0 qpβ 1 px 0 q ´εqs . Combining ( 10), ( 11), ( 13), ( 16), Theorem 1 follows.

Proof of Theorem 2

We use the decomposition p θ β,p px 0 q θ β,p px 0 q " # ˆk np ˙βrp p k np q βγ 1 px 0 q θ β, k n px 0 q θ β,p px 0 q ´1, .

-

. ( 17
)
We will study the three terms pT piq β,n ´1q, i " 1, 2, 3, separately. Using the assumption ln k{pnpq ?

´1)

β γ 1 px 0 q f X px 0 q "ż 1 0 W pz, 8q 1 z dz ´W p1, 8q

 . ( 18
)
The asymptotic behavior of the term pT p2q β,n ´1q has been already established in Theorem 1. Now, according to (4), we have T p3q β,n ´1 " ˜θβ,k{n px 0 q{rU 1 pn{k|x 0 qs β θ β,p px 0 q{rU 1 p1{p|x 0 qs β ´1¸r U 1 pn{k|x 0 qs β p k np q βγ 1 px 0 q rU 1 p1{p|x 0 qs β `rU 1 pn{k|x 0 qs β p k np q βγ 1 px 0 q rU 1 p1{p|x 0 qs β ´1 " ˜θβ,k{n px 0 q{rU 1 pn{k|x 0 qs β θ β,p px 0 q{rU 1 p1{p|x 0 qs β ´1¸r U 1 pn{k|x 0 qs β p k np q βγ 1 px 0 q rU 1 p1{p|x 0 qs β `" 1 `a1 pn{k|x 0 q 1 `a1 p1{p|x 0 q  β ´1 " ˜θβ,k{n px 0 q{rU 1 pn{k|x 0 qs β R n{k `p s n p1, z|x 0 q, 1 ˇˇx 0 ˘dz ´γ1 px 0 q " ´ż 8 0 R ˜"1 `z´γ 1 px 0 q β  ´1 γ 1 px 0 q , 1 ˇˇx 0 ¸dz ´γ1 px 0 q ´ż 8 0 " R n{k `p s n p1, z|x 0 q, 1 ˇˇx 0 ˘´R `p s n p1, z|x 0 q, 1 ˇˇx 0 ˘‰ dz ´γ1 px 0 q ´ż 8 0 « R `p s n p1, z|x 0 q, 1 ˇˇx 0 ˘´R ˜"1 `z´γ 1 px 0 q β  ´1 γ 1 px 0 q , 1 ˇˇx 0 ¸ff dz ´γ1 px 0 q ": ´ż 8 0 R ˜"1 `z´γ 1 px 0 q β  ´1 γ 1 px 0 q , 1 ˇˇx 0 ¸dz ´γ1 px 0 q `T p4q β,n `T p5q β,n .

Note that T p4q β,n is similar to T 3,1,n and hence T p4q β,n " op1{ a kh d n q. Further, T p5q β,n is similar to T 3,2,n and hence, for a sequence T n as used for T 3,2,n , we have T p5q β,n " op1{ a kh d n q. Thus θ β,k{n px 0 q rU 1 pn{k|x 0 qs β " ´ż 8 0 R ˜"1 `z´γ 1 px 0 q β  ´1 γ 1 px 0 q , 1 ˇˇx 0 ¸dz ´γ1 px 0 q `o ˜1 a kh d n ¸.

Similarly, we have θ β,p px 0 q rU 1 p1{p|x 0 qs β " ´ż 8 0 R ˜"1 `z´γ 1 px 0 q β  ´1 γ 1 px 0 q , 1 ˇˇx 0 ¸dz ´γ1 px 0 q ´ż 8 0 " R 1{p ˆ1 p

" F 1 ˆ"1 `z´γ 1 px 0 q β  U 1 ˆ1 p ˇˇx 0 ˙ˇˇx 0 ˙ , 1 ˇˇx 0 Ṙ ˆ1 p " F 1 ˆ"1 `z´γ 1 px 0 q β  U 1 ˆ1 p ˇˇx 0 ˙ˇˇx 0 ˙ , 1 ˇˇx 0 ˙ dz ´γ1 px 0 q ´ż 8 0 " R ˆ1 p " F 1 ˆ"1 `z´γ 1 px 0 q β  U 1 ˆ1 p ˇˇx 0 ˙ˇˇx 0 ˙ , 1 ˇˇx 0 Ṙ ˜"1
`z´γ 1 px 0 q β  ´1 γ 1 px 0 q , 1 ˇˇx 0 ¸ff dz ´γ1 px 0 q ": ´ż 8 0 R ˜"1 `z´γ 1 px 0 q β  ´1 γ 1 px 0 q , 1 ˇˇx 0 ¸dz ´γ1 px 0 q `T p6q β,n `T p7q β,n .

Clearly T p6q β,n " Opp ´τ q " o ˆ1 ? ´"1 `z´γ 1 px 0 q β  ´1 γ 1 px 0 q ˇˇˇˇd z ´γ1 px 0 q `2 sup under the same conditions ( 14) and ( 15) on the sequence T n . Combining all these results together yield

T p3q β,n ´1 " o ˜1 a kh d n ¸. (19)
Finally, decomposition (17) combined with Theorem 1, ( 18) and ( 19) achieves the proof of Theorem 2.

Figure 1 :

 1 Figure1: Model 1. Boxplots of p θ 1,p px 0 q for n " 500 (top row) and n " 1 000 (bottom row), and p " 1{n (left) and p " 1{p2nq (right), along with θ 1,p px 0 q (red curve).

Figure 2 :

 2 Figure2: Model 1. Boxplots of p θ 1.5,p px 0 q for n " 500 (top row) and n " 1 000 (bottom row), and p " 1{n (left) and p " 1{p2nq (right), along with θ 1.5,p px 0 q (red curve).

Figure 3 :

 3 Figure 3: Model 2. Boxplots of pθ 1,p px 0 q for n " 500 (top row) and n " 1 000 (bottom row), and p " 1{n (left) and p " 1{p2nq (right), along with θ 1,p px 0 q (red curve).

Figure 4 :

 4 Figure 4: Model 2. Boxplots of p θ 1.25,p px 0 q for n " 500 (top row) and n " 1 000 (bottom row), and p " 1{n (left) and p " 1{p2nq (right), along with θ 1.25,p px 0 q (red curve).

Figure 5 :

 5 Figure 5: Model 3. Boxplots of pθ 1,p px 0 q for n " 500 (top row) and n " 1 000 (bottom row), and p " 1{n (left) and p " 1{p2nq (right), along with θ 1,p px 0 q (red curve).

Figure 6 :

 6 Figure 6: Model 3. Boxplots of p θ 1.2,p px 0 q for n " 500 (top row) and n " 1 (bottom row), and p " 1{n (left) and p " 1{p2nq (right), along with θ 1.2,p px 0 q (red curve).

Figure 7 :

 7 Figure 7: γ 1 px 0 q as a function of x 0 for Model 2 (left) and Model (right).

1

 1 and p F n,2 are given in (3). The values of k and k 1 are determined by using a data driven procedure proposed in Goegebeur et al. (2019), after which we estimate r by b k 1

Figure 8 :

 8 Figure 8: Vehicle insurance dataset. Scatterplot for the total claim amount versus the customer's lifetime value.

b kh d n ˇˇˇż 8 0 " R `r s n py 1 ," 1 `u´γ 1 px 0 q β ˙´1 γ 1 px 0 q , y 2 ˇˇx 0 ¸ff du ´γ1 px 0 q ˇˇˇď b kh d n ˇˇˇż δn 0 " R `r s n py 1 ," 1 `u´γ 1 px 0 q β ˙´1 γ 1

 0110111 u|x 0 ´hn vq, r t n py 2 |x 0 ´hn vq|x 0 Ȓ ˜ˆy 1 u|x 0 ´hn vq, r t n py 2 |x 0 ´hn vq|x 0 Ȓ ˜ˆy 1 px 0 q

Finally, sup y 1 ,y 2 ,2s ˜´ż 8 δ

 128 Pr1{2

  6 and the proof of Theorem 3, respectively, in Goegebeur et al. (2021a).

´ż 8 0R ˆp u ´1 γ 1 px 0 q n, 1 " 1 `u´γ 1 px 0 q β ı ´1 γ 1 px 0 q , p e n, 2 1 `u´γ 1 px 0 q β ı ´1 γ 1 px 0 q , 1 ˇˇx0 Ṙ ¨p u ´1 γ 1 px 0 q n, 1 p e n, 2 " 1 `u´γ 1 px 0 q β ı ´1 γ 1 px 0 q , 1 ˇˇx0 'fi fl du ´γ1px0q `p1 ´p e n,2 q ż 8 0R ˆ" 1 `u´γ 1 px 0 q β ı ´1 γ 1 px 0 q , 1 ´H ¨p u ´1 γ 1 px 0 q n, 1 p e n, 2 'fifl `bkh d n p1 ´p e n, 2 q

 81121121´γ1px0q811122 Hp1q `oP p1q,

T 2,n " " b kh d n pp e n,2 ´1q ´bkh d n ˆp u ´1 γ 1 px 0 q n, 1 ´1˙ H 1 p p ξ n q ´bkh d n pp e n, 2 ´1q´1 γ 1 px 0 q n, 1 p e n, 2

 1212 Hp1q `oP p1q, where p ξ n is an intermediate random value between 1 and p u

|T 3 1 1 ^1 ˆˇˇˇż 8 0|T 3

 31183 ă8, 1 2 ďy 2 ď2 |R n{k py 1 , y 2 |xq ´Rpy 1 , y 2 |xq| y κrpp s n p1, u|x 0 qq κ ^1s du ´γ1 px 0 q ˇˇ" O ˆbkh d n ´n k ¯τ ˙" op1q, by our assumptions on the sequence k and since κ ą βγ 1 px 0 q. Then, for T 3,2,n we use the decomposition

|T 3

 3

for a sequence T n such that b kh d n T ´γ1 px 0k |x 0 ¯|x 0 ¯ˇˇT 1´γ 1

 01 px 0 qrβ´β 1 px 0 q`εs β n ÝÑ 0.

From ( 12 )

 12 , we have the following further conditions ¯τ ÝÑ 0, leading to the additional bounds on ∆ ∆ ą max ˆα d `2pη R ^ηf X q , α `2p1 ´αqτ q d ˙.

γ 1 ,k 1 q βγ 1 ,

 111 px 0 q´γ 1 px 0 qs px 0 q θ β, from which we deduce that p θ β,p px 0 q θ β,p px 0 q ´1 " # ˆk np ˙βrp γ 1,k 1 px 0 q´γ 1 px 0 qs ´1+ T

θ β,p px 0 q{rU 1 a kh d n |δ 1 pU 1 pn{k|x 0 q|x 0 q| Ñ 0 and a kh d n |δ 1

 11 p1{p|x 0 qs β ´1¸˜1 `pU 1 p1{p|x 0 q|x 0 q| Ñ 0 under our assumptions. Remark now that θ β,k{n px 0 q rU 1 pn{k|x 0 qs β " ´ż 8 0

  by our assumptions on k and p.Finally, similarly as for T p5q β,n , for T n a sequence that tends to infinity, we have, for n large,

  1 `0.1x. This model satisfies pSq with Rpy 1 , y 2 |xq " y 1 `y2 ´ay 2

	1	`y2 2 , τ " ´1 and β " 2 (see, e.g., Cai
	et al. 2015, in the context without covariates).

  1 px 0 qγ 1 px 0 qp1 ´αq d , α d `2pη A 2 ^ηγ 2 qr1 ´βγ 1 px 0 qs , α ´2β 2 px 0 qγ 2 px 0 qp1 ´αqp1 ´βγ 1 px 0 qq d `2pη ε 2 ^ηB 2 qr1 ´βγ 1 px 0 qs R ´r s n py 1 , u|x 0 ´hn vq, r t n py 2 |x 0 ´hn vq ˇˇx 0

	one has that
	sup vPS K	y 1 ,y 2 Pr1{2,2s sup	0 n ˇˇˇż kh d b 8	"		R
			˜ˆy 1	"	1	β `u´γ 1 px 0 q	˙´1 γ 1 px 0 q
						˙ă ∆ ă	α d

  1`1{γ 1 px 0 q s,y 2 Pr1{2,2s ˇˇakh d n pT n py 1 , y 2 |x 0 q ´EpT n py 1 , y 2 |x 0 qqq ´W py 1 , y 2 q ˇy rp s n py 1 , u|x 0 qs η du ´γ1 px 0 q ˙ď C

					η
					1
		ˆsup	ˆ´ż 8	rp s n py 1 , u|x 0 qs η du ´γ1 px 0 q	˙.
		y 1 Pr1{2,2s	0
	Using Proposition B.1.10 in de Haan and Ferreira (2006) one obtains that for n ą n 1
		ˆ´ż 8		ż 8
	sup
	y 1 Pr1{2,2s	0		0

  1 in Goegebeur et al. (2021a), see also Theorem 3, we deduce that T 5 " o P p1q. Now consider T 6 . We have for any δ ą 0 T 6 ď sup y 1 ,y 2 Pr1{2,2s,uďδ|W pp s n py 1 , u|x 0 q, y 2 q| rp s n py 1 , u|x 0 qs η sup rp s n py 1 , u|x 0 qs η du ´γ1 px 0 q

		ˆ´ż δ
	y 1 Pr1{2,2s	0

ṡup y 1 ,y 2 Pr1{2,2s,uďδ

  1`1{γ 1 px 0 q s,y 2 Pr1{2,2s

	|W py 1 , y 2 q| 1 y η	,
	these two bounds being finite with probability 1 according to Lemma 2 in Cai et al. (2015).
	Also, remark that
	sup
	y 1 Pr1{2,2s

  2,n and T 3,n . Concerning T 1,n , by Proposition 1 and following the lines of proof of Theorem 3 in Goegebeur et al. (2021a) we obtain

  r t n py 2 |x 0 ´hn vq|x 0 ´hn v Ȓ `r s n py 1 , u|x 0 ´hn vq, r t n py 2 |x 0 ´hn vq|x 0 ´hn v ˘‰ f X px 0 ´hn vqdvdu ´γ1 px 0 q

	`ż 8	ż	Kpvq	"	R `r s
	0	S K

n py 1 , u|x 0 ´hn vq, r t n py 2 |x 0 ´hn vq|x 0 ´hn v Ȓ `r s n py 1 , u|x 0 ´hn vq, r t n py 2 |x 0

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.

[25] Yao, Q.

(1999