Yuri Goegebeur

Armelle Guillou

Nguyen Khanh

Le Ho

Jing Qin

A Weissman-type estimator of the conditional marginal expected shortfall

Keywords: Conditional marginal expected shortfall, extrapolation, Pareto-type distribution

The marginal expected shortfall is an important risk measure in finance and actuarial science, which has been extended recently to the case where the random variables of main interest are observed together with a covariate. This leads to the concept of conditional marginal expected shortfall 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 proposed estimator is evaluated with a simulation experiment, and the practical applicability is illustrated on vehicle insurance customer data.

Introduction

A central topic in actuarial science and finance is the quantification of the risk of a loss variable. This is done by risk measures, the most basic among them is the Value-at-Risk (VaR), defined as the p´quantile of a loss variable Y :

Qppq :" infty : F Y pyq ě pu, p P p0, 1q, where F Y denotes the distribution function of Y . We refer to [START_REF] Jorion | Value at risk: the new benchmark for managing financial risk[END_REF] for a review. The main drawbacks of VaR are that it does not take the loss above this p´quantile into consideration and it is not a coherent risk measure [START_REF] Artzner | Coherent measures of risk[END_REF]. Recently, the conditional tail expectation (CTE), defined as CT Eppq " EpY |Y ą Qppqq, p P p0, 1q, became a popular alternative to Value-at-Risk. It is more conservative than VaR and it is also a coherent risk measure. This measure has been extensively studied in the literature, see, e.g., 1 [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]. The conditional tail expectation has also been extended to the multivariate context, leading to the concept of the marginal expected shortfall (MES). For a pair of risk factors pY p1q , Y p2q q, the marginal expected shortfall is defined as θ p " EpY p1q |Y p2q ą Q 2 p1 ´pqq, p P p0, 1q, where Q 2 denotes the quantile function of risk factor Y p2q . This measure was introduced by [START_REF] Acharya | Measuring systemic risk[END_REF], 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, where n is the sample size, leading to extrapolations outside the data range. We also refer to [START_REF] Landsman | Tail conditional expectations for elliptical distributions[END_REF], [START_REF] Cai | Conditional tail expectations for multivariate phase-type distributions[END_REF], [START_REF] Bargès | TVar-based capital allocation with copulas[END_REF], [START_REF] Cousin | On multivariate extensions of value-at-risk[END_REF][START_REF] Cousin | On multivariate extensions of value-at-risk[END_REF][START_REF] Di Bernardino | Estimation of the multivariate conditional tail expectation for extreme risk levels: Illustration on environmental data sets[END_REF], Das andFasen-Hartmann (2018, 2019).

Recently, [START_REF] Goegebeur | Conditional marginal expected shortfall[END_REF] have considered the estimation of the marginal expected shortfall, but this time in case where the random variables of main interest pY p1q , Y p2q q are recorded together with a random covariate X P R d . This leads to the concept of conditional marginal expected shortfall, given X " x 0 , and defined as

θ p px 0 q " E " Y p1q ˇˇY p2q ě Q Y p2q ´1 ´pˇˇˇx 0 ¯; X " x 0 ı .
Note that in the financial and actuarial setting where risk measures and in particular MES have been introduced, one is often interested in positive risk factors. Thus, in the sequel, we consider the case where Y p1q and Y p2q are positive. The extension to the case of a real-valued Y p1q is complicated and this topic is moreover also an open problem in the much simpler case where there are no covariates, but it will lead to further investigations.

In the sequel, we will denote by F j p¨|xq the continuous conditional distribution function of Y pjq , j " 1, 2, given X " x, and use the notation F j p¨|xq for the conditional survival function and U j p¨|xq for the associated tail quantile function defined as U j p¨|xq " infty : F j py|xq ě 1´1{¨u. Also, we will define by f X the density function of the covariate X and by x 0 a reference position such that x 0 P IntpS X q, the interior of the support S X Ă R d of f X , which is assumed to be non-empty. Considering pY p1q i , Y p2q i , X i q, i " 1, . . . , n, independent copies of pY p1q , Y p2q , Xq, [START_REF] Goegebeur | Conditional marginal expected shortfall[END_REF] proposed for the situation where the conditional distribution functions of Y p1q and Y p2q given X " x are of Pareto-type the following nonparametric estimator for θ k{n px 0 q θ k n px 0 q :"

1 k ř n i"1 K hn px 0 ´Xi qY p1q i 1l tY p2q i ě p U 2 pn{k|x 0 qu p f n px 0 q ,
where K hn p.q :" Kp.{h n q{h d n , with K a joint density function on R d , k an intermediate sequence such that k Ñ 8 with k{n Ñ 0, h n a positive non-random sequence of bandwidths with h n Ñ 0 if n Ñ 8, 1l A the indicator function on the event A, and p f n px 0 q :" p1{nq ř n i"1 K hn px 0 ´Xi q is a classical kernel density estimator. Here, p U 2 p.|x 0 q is an estimator for U 2 p.|x 0 q, defined as p U 2 p.|x 0 q :" infty : p F n,2 py|x 0 q ě 1 ´1{.u where p F n,2 py|x 0 q :" 1 n ř n i"1 K hn px 0 ´Xi q1l

tY p2q i ďyu p f n px 0 q . ( 1
)
The asymptotic behavior of θ k{n px 0 q has been established by [START_REF] Goegebeur | Conditional marginal expected shortfall[END_REF] and is recalled in Theorem 5.1 in Section 5. Due to the conditions k, n Ñ 8 with k{n Ñ 0, the Y p2q -quantile is intermediate, and thus the estimator θ k{n px 0 q cannot be used for extrapolation outside the Y p2q -data range. Extrapolation is relevant for practical data analysis, since often we want to go beyond the range of the available data. With a classical empirical estimator like θ p px 0 q where p ă 1{n this is not possible, as it would always lead to the trivial estimate of zero. The aim of this paper is to solve this issue and to define a new estimator which allows extrapolation and thus which is valid for p ă 1{n. This requires the intermediate estimator θ k{n px 0 q but also an estimator for the conditional extreme value index of the distribution of Y p1q given X " x 0 . Since we need to consider θ k{n px 0 q and the conditional tail index estimator jointly, we introduce a conditional tail index estimator, and we analyse it in terms of an empirical process, related to the process needed in the analysis of θ k{n px 0 q. Note that in [START_REF] Girard | Extreme conditional expectile estimation in heavy-tailed heteroscedastic regression models[END_REF] a tail index estimator is proposed in a location-scale model, and it is analysed with a tail empirical process of residuals.

The Value-at-Risk and conditional tail expectation mentioned above have also been studied in an extreme value framework with random covariates. As for the estimation of extreme conditional quantiles we refer to [START_REF] Daouia | Kernel estimators of extreme level curves[END_REF][START_REF] Daouia | On kernel smoothing for extremal quantile regression[END_REF]. In El [START_REF] El Methni | Non-parametric estimation of extreme risk measures from conditional heavy-tailed distributions[END_REF], for the framework of heavy-tailed distributions, the conditional tail expectation was generalised to the conditional tail moment, and estimators were introduced for the situation where the variable of main interest Y was observed together with a random covariate X. This work was extended to the general max-domain of attraction in El [START_REF] El Methni | Kernel estimation of extreme regression risk measures[END_REF].

The remainder of the paper is organized as follows. In Section 2, we introduce our estimator for the conditional marginal expected shortfall allowing extrapolation and we establish its main asymptotic properties. The efficiency of our estimator is examined with a small simulation study in Section 3. Finally, in Section 4 we illustrate the performance of our estimator on vehicle insurance customer data. Some preliminary results are given in Section 5, whereas the proofs of the main results are postponed to Section 6.

Estimator and asymptotic properties

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.

Clearly, Assumption pDq implies 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 .

Now, to estimate θ p px 0 q it is required to impose an assumption on the right-hand upper tail dependence of pY p1q , Y p2q q, conditional on a value of the covariate 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.

Assuming pDq with γ 1 px 0 q ă 1 and pRq, one can show (see [START_REF] Cai | Estimation of the marginal expected shortfall: the mean when a related variable is extreme[END_REF], Proposition 1) that lim pÑ0 θ p px 0 q U 1 p1{p|x 0 q " ´ż 8 0 Rps, 1|x 0 qds ´γ1 px 0 q , from which the following approximation can be deduced

θ p px 0 q " U 1 p1{p|x 0 q U 1 pn{k|x 0 q θ k n px 0 q " ˆk np ˙γ1 px 0 q θ k n px 0 q.
Thus, to estimate θ p px 0 q, we need first to estimate γ 1 px 0 q. We propose the following 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 , (3)
based on an intermediate sequence k 1 such that k 1 Ñ 8 with k 1 {n Ñ 0. This sequence may be different from k, the one used in the estimator θ k n px 0 q, but the two sequences are linked together (see Theorem 2.2 below). On the contrary, for convenience, we use the same bandwidth h n and kernel K for both the estimation of γ 1 px 0 q and θ p px 0 q. Note that p γ 1,k 1 px 0 q is a local version of the Hill estimator [START_REF] Hill | A simple general approach to inference about the tail of a distribution[END_REF] from the univariate extreme value context. The extreme value literature contains several alternative estimators for the conditional tail index. We refer to [START_REF] Gardes | A moving window approach for nonparametric estimation of the conditional tail index[END_REF], [START_REF] Daouia | Kernel estimators of extreme level curves[END_REF][START_REF] Dierckx | Local robust and asymptotically unbiased estimation of conditional Pareto-type tails[END_REF] and [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails -the random covariate case[END_REF] for conditional tail index estimators in the framework of Pareto-type distributions, and to [START_REF] Daouia | On kernel smoothing for extremal quantile regression[END_REF], [START_REF] Stupfler | A moment estimator for the conditional extreme-value index[END_REF] and Goegebeur et al. (2014a) for conditional tail index estimators that work in the broader general max-domain of attraction. Now, we are able to define a Weissman-type estimator for θ p px 0 q, given by

p θ p px 0 q " ˆk np ˙p γ 1,k 1 px 0 q θ k n px 0 q.
Our aim in this paper is to establish the asymptotic behavior of p θ p px 0 q, which requires the asymptotic behavior of p γ 1,k 1 px 0 q in terms of the process on which the estimator θ k n px 0 q is based on (see Theorem 5.1 in Section 5). To reach this goal, some assumptions due to the regression context are required. In particular, 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 β ą γ 1 px 0 q, 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 β 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 , and
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.

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 }.}.

Our first aim, now, is to show the weak convergence, denoted , of the process based on p γ 1,k 1 px 0 q, but in terms of the same process as the one used in Theorem 2.3 from [START_REF] Goegebeur | Conditional marginal expected shortfall[END_REF].

Theorem 2.1 Assume pDq, pHq, pKq, x 0 P IntpS X q with f X px 0 q ą 0, and y Ñ F 1 py|x 0 q, is strictly increasing. Consider sequences k 1 Ñ 8 and h n Ñ 0 as n Ñ 8, in such a way that k

1 {n Ñ 0, k 1 h d n Ñ 8, h ηε 1 n ln n{k 1 Ñ 0, a k 1 h d n h η f X ^ηA 1 n Ñ 0, a k 1 h d n h ηγ 1 n ln n{k 1 Ñ 0, a k 1 h d n |δ 1 pU 1 pn{k 1 |x 0 q|x 0 q| Ñ 0. Then we have, b k 1 h d n pp γ 1,k 1 px 0 q ´γ1 px 0 qq γ 1 px 0 q f X px 0 q "ż 1 0 W pz, 8q 1 z dz ´W p1, 8q  ,
where W pz, 8q is a zero centered Gaussian process with covariance function

EpW pz, 8qW pz, 8qq " }K} 2 2 f X px 0 q pz ^zq ,

with }K} 2 :" b ş R d K 2 puqdu.
Note that the variance of the limiting distribution of p γ 1,k 1 px 0 q, after normalization, is given by γ 2 1 px 0 q}K} 2 2 {f X px 0 q, compared to an asymptotic variance of γ 2 1 for the Hill estimator in the univariate context.

Before stating the weak convergence of p θ p px 0 q, we need to introduce a second order condition, usual in the extreme value context.

Assumption pSq. There exist β ą γ 1 px 0 q 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 β 1 ^1 " Opt τ q,
for some r ą 0.

Our final result is now the following.

Theorem 2.2 Assume pDq, pHq, pKq, pSq with x Ñ Rpy 1 , y 2 |xq being a continuous function, 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 `2γ 1 px 0 qpη R ^ηA 1 ^ηγ 1 q , α d `2p1 ´γ1 px 0 qqpη A 2 ^ηγ 2 ^ηB 2 ^ηε 2 ^ηf X q , α d ´2p1 ´αqγ 2 1 px 0 qβ 1 px 0 q d `dpβ 1 px 0 q `εqγ 1 px 0 q , α ´2p1 ´αqpγ 1 px 0 q ^pβ 2 px 0 qγ 2 px 0 qq ^p´τ qq d , α 1 d `2 pη f X ^ηA 1 ^ηγ 1 q , α 1 ´2p1 ´α1 qγ 1 px 0 qβ 1 px 0 q d ¯ă ∆ ă α d .
Then, for γ 1 px 0 q ă 1{2 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 p px 0 q θ p px 0 q ´1¸ minpr, 1q γ 1 px 0 q f X px 0 q ˆż 1 0 W py, 8q 1 y dy ´W p1, 8q ṁin ˆ1, 1 r ˙#´p1 ´γ1 px 0 qq W p8, 1q f X px 0 q `1 f X px 0 q
ş 8 0 W py, 1qdy ´γ1 px 0 q ş 8 0 Rpy, 1|x 0 qdy ´γ1 px 0 q + , 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, W py, 8q is the limiting process of Theorem 2.1, and W p8, yq is a zero centered Gaussian process with covariance function E pW p8, yqW p8, yqq " }K} 2 2 f X px 0 qpy ^yq.

The variance of the limiting random variable in Theorem 2.2, denoted W, is given by

VarpWq " }K} 2 2 f X px 0 q # pminpr, 1qq 2 γ 2 1 px 0 q `ˆmin ˆ1, 1 r ˙˙2 " γ 2 1 px 0 q ´1 ´c2 ż 8 0
Rps, 1|x 0 qds ´2γ 1 px 0 q  `2 min ˆr, 1 r ˙γ1 px 0 q « p1 ´γ1 px 0 q `cq Rp1, 1|x 0 q ´ż 1 0 ´1 ´γ1 px 0 q `cs ´γ1 px 0 q p1 ´γ1 px 0 q ´γ1 px 0 q ln sq ¯Rps, 1|x 0 q s ds ff+ ,

where c :" p ş 8 0 Rps, 1|x 0 qds ´γ1 px 0 q q ´1.

If one takes k " k 1 or k{k 1 Ñ c P p0, 8q in Theorem 2.2, then the rate of convergence of θ k{n px 0 q and p γ 1,k 1 px 0 q are the same. If one assumes additionally that np " opkq, then the limiting distribution of a kh d n { lnpk{pnpqqp p θ p px 0 q{θ p px 0 q ´1q is essentially that of a k 1 h d n pp γ 1,k 1 px 0 q γ1 px 0 qq. In order words, the extrapolated estimator for CMES inherits the limiting distribution of the estimator for γ 1 px 0 q. A similar fact is observed in the estimation of extreme quantiles (see, e.g., Theorem 4.3.8 in de [START_REF] De Haan | Extreme value theory, an introduction[END_REF] and in the estimation of extreme conditional quantiles (e.g., [START_REF] Daouia | Kernel estimators of extreme level curves[END_REF], where the limiting distribution of the (conditional) extreme quantile estimator is that of the estimator of the (conditional) extreme value index. The result of Theorem 2.2 is more general, as, depending on the value of r, one can have as limiting distribution a linear combination of the limiting distributions of a kh d n pθ k{n px 0 q{θ k{n px 0 q ´1q and a k 1 h d n pp γ 1,k 1 px 0 q ´γ1 px 0 qq.

Simulation experiment

In this section, we illustrate the finite-sample performance of our conditional marginal expected shortfall estimator with a simulation experiment. To this aim, we consider the three models already used in [START_REF] Goegebeur | Conditional marginal expected shortfall[END_REF], but this time in the case where p ă 1{n, i.e., allowing extrapolation outside the Y p2q ´data range. These models are the following:

Model 1: The conditional logistic copula model, defined as 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)

In this model, X is uniformly distributed on the interval r2, 10s, and the marginal distribution functions of Y p1q and Y p2q are Fréchet distributions:

F j pyq " e ´y´1{γ j , y ą 0, j " 1, 2. We set γ 1 " 0.25 and γ 2 " 0.5. It can be shown that this model satisfies Assumption pSq with Rpy 1 , y 2 |xq " y 1 `y2 ´py x 1 `yx 2 q 1{x , τ " ´1 and β " 1 ´ε for some small ε ą 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 .
Here, X is uniformly distributed on r0, 1s and γ 1 pxq " 0.4 r0.1 `sinpπxqs " 1.1 ´0.5e ´64px´0.5q 2 ı , γ 2 pxq " 0.1 `0.1x.

Again, Assumption pSq is satisfied for this model with Rpy 1 , y 2 |xq " y 1 `y2 ´ay 2 1 `y2 2 , τ " ´1 and β " 2 (see, e.g., [START_REF] Cai | Estimation of the marginal expected shortfall: the mean when a related variable is extreme[END_REF], in the context without covariates).

Model 3: We consider again the conditional logistic copula model defined in (5) but this time with conditional Burr distributions for the marginal distribution functions of Y p1q and Y p2q , i.e., F j py|xq " 1 ´ˆβ j β j `yτ j pxq ˙λj , y ą 0; β j , λ j , τ j pxq ą 0, j " 1, 2. We set β 1 " β 2 " 1, λ 1 " 1, λ 2 " 0.5, and τ 1 pxq " 2e 0.2x , τ 2 pxq " 8{ sinp0.3xq.

Similarly to Model 1, this model satisfies pSq.

Note that for all these models, Assumption pDq is satisfied since all the marginal conditional distributions are standard examples from this class of heavy-tailed distributions (see, e.g., [START_REF] Beirlant | Second-order refined peaks-over-threshold modelling for heavy-tailed distributions[END_REF], Table 1), and Assumption pHq also holds.

We simulate 500 datasets of size n " 500 and 1 000 from each model. For each sample, we compute p θ p px 0 q for two different values of p: 1{p2nq and 1{p5nq, and three different sets of values of the covariate position x 0 : t3, 5, 7u for Model 1 andModel 3, andt0.3, 0.5, 0.8u for Model 2. Concerning the kernel function K, we use the bi-quadratic function

Kpxq " 15 16 p1 ´x2 q 2 1l txPr´1,1su ,
for both the estimation of p γ 1,k 1 px 0 q and θ k{n px 0 q. This kernel function clearly satisfies Assumption pKq. To compute these estimators, we need also to select 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 , ( 6
)
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 .
Concerning the choice of the sequence k 1 in the estimation of γ 1 px 0 q, a graphical assessment is used. The retained value of k 1 corresponds to the smallest value of k after which the median of the estimates p γ 1,k px 0 q over the 500 replications exibit a stable part. This choice is common in extreme value theory where we search at a plateau of an extreme value estimator as a function of the intermediate sequence.

Figures 1 to 6 display the boxplots of the ratios between the estimates p θ p px 0 q and the true value θ p px 0 q based on the 500 replications for the different values of x 0 (corresponding to the rows) and the two sample sizes (corresponding to the columns: n " 500 on the left, n " 1 000 on the right). Figures 1 and2 correspond to Model 1 for p " 1{p2nq and 1{p5nq, respectively. Similarly, Figures 3 and4 correspond to Model 2, and Figures 5 and6 to Model 3.

Based on these simulations, we can draw the following conclusions:

• Our estimator p θ p px 0 q performs quite well in all the situations, although its efficiency depends obviously on the model and also on the covariate position. This is expected because as is clear from our models, the marginal distributions in Model 1 do not depend on the covariates but the dependence structure does. On the contrary, the dependence structure in Model 2 does not depend on the covariate but the marginal distributions depend on x 0 , and in Model 3 both of them depend on x 0 . Thus, Model 1 is less challenging than the two other models;

• Note that the estimation in Model 2 is difficult and depends a lot on the value of the covariate. This can be explained by the fact that the plot of γ 1 pxq as a function of x exhibits two local maxima, one of which being close to 0.3, and a local minimum at 0.5. Thus, we get an underestimation of θ p px 0 q near the local maxima and an overestimation of θ p px 0 q near the local minima, due to the local nature of the estimation. Outside these neighborhoods, the estimation is without bias. This is the case for x 0 " 0.8. Note also that sometimes a local bandwidth instead of a global one as in ( 6) can give better results, especially at covariate positions where the function γ 1 pxq changes quickly;

• As expected, the smaller p is, the more difficult is the estimation, due to an important extrapolation outside the Y p2q ´data range. This results in an increase of the variability of the estimates. Note that the values p " 1{p2nq and p " 1{p5nq correspond to quite severe extrapolations since the estimation is done locally, and the local number of observations is much smaller than n.

Next, we investigate in Table 1 the coverage probabilities of the pointwise 95% confidence intervals for θ p px 0 q, based on a log-scale version of Theorem 2.2, namely for

min ˜bkh d n , a k 1 h d n ln k{pnpq ¸ln p θ p px 0 q θ p px 0 q ,
which has the same limiting distribution as in Theorem 2.2, and this for the three models with their different values of x 0 , and three values for p: 1{n, 1{p2nq and 1{p5nq. These confidence intervals are given by

» - - - - - p θ p px 0 q exp # Φ ´1 `1 ´α 2 ˘b { VarpWq an + , p θ p px 0 q exp # ´Φ´1 `1 ´α 2 ˘b { VarpWq an + fi ffi ffi ffi ffi fl , (7)
where

a n :" minp a kh d n , a k 1 h d n {pln k{pnpqq, Φ ´1 denotes

the standard normal quantile function and {

VarpWq is an estimate for the asymptotic variance given in (4), obtained by using the local Hill estimate (3) for γ 1 px 0 q and the following estimate for Rpy 1 , y 2 |x 0 q:

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 , ( 8
)
where p F n,1 is a kernel estimator for F 1 , of the same form as p F n,2 given in (1). Note that this estimator (8) can be viewed as an adjusted version of the estimator proposed in the context of estimation of the conditional stable tail dependence function by [START_REF] Escobar-Bach | Local estimation of the conditional stable tail dependence function[END_REF] 1: Empirical coverage probabilities of 95% confidence intervals for θ p px 0 q based on 500 simulated datasets of size n " 1000 with h " h cv defined in (6) (with h " h cv {2).

Remark also that we use here the log-scale version of Theorem 2.2 as suggested by [START_REF] Drees | Extreme quantile estimation for dependent data, with applications to finance[END_REF] since it improves the coverage probabilities. This can be explained by the fact that the normal approximation of logp p θ p px 0 q{θ p px 0 qq is more accurate than the one of p θ p px 0 q{θ p px 0 q ´1, since by definition of p θ p px 0 q, the log-transform yields to a linear function of p γ 1,k 1 px 0 q, which distribution is well approximated by a normal distribution. For the construction of the confidence intervals, k and k 1 were selected in a data driven way, by using a stability criterion as in [START_REF] Goegebeur | Bias-corrected estimation for conditional Paretotype distributions with random right censoring[END_REF]. Concerning h, the global bandwidth h cv defined in ( 6) is first used. Note that the practical implementation of the confidence intervals based on Theorem 2.2 also requires the value r. The latter is taken as b k k 1 ln k np , for the data-driven choices of k and k 1 . Overall the confidence intervals have reasonably good coverage probabilities, if one takes into account the fact that the asymptotic variance given in (4) has a complicated form with several integrals that depend on Rpy 1 , y 2 |x 0 q, which needs to be replaced by an estimator. As expected, the coverage probabilities of the asymptotic confidence intervals depend on the model and the covariate positions: the positions where θ p px 0 q is estimated well (with no/little bias) give good coverage probabilities, and those with bias lead typically to smaller coverage probabilities. To improve them, a solution is to use a local bandwidth leading to a smaller value of h. This is illustrated in Table 1 where a heuristic value h " h cv {2 is also used. As is clear from that table, the coverage probabilities improve a lot, being closer to the nominal level, in the cases where the estimation is biased with a global bandwidth. From Table 1, we can also remark that the coverage probabilities are not too much sensitive on the value of p. Alternative methods, such as empirical likelihood or bootstrap approaches, could be also investigated in future research to see their impact on coverage probabilities.

Application to vehicle insurance data

In this section, we illustrate our method on the Vehicle Insurance Customer Data, available at https://www.kaggle.com/ranja7/vehicle-insurance-customer-data, which contains socio-q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 10 25 50 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 200 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 10 25 50 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 200 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 10 25 50 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 200 0.0 0.5 1.0 1.5 2.0 2.5

k

Figure 1: Boxplots of p θ p px 0 q{θ p px 0 q for Model 1 based on 500 replications of sample sizes n " 500 (left column) and n " 1 000 (right column) for p " 1{p2nq. From top to bottom we have x 0 " 3, x 0 " 5 and x 0 " 7. The values of k are taken as 2%, 5%, 10% and 20% of n. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 10 25 50 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 200 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 10 25 50 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 200 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 10 25 50 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 θ p px 0 q{θ p px 0 q for Model 1 based on 500 replications of sample sizes n " 500 (left column) and n " 1 000 (right column) for p " 1{p5nq. From top to bottom we have x 0 " 3, x 0 " 5 and x 0 " 7. The values of k are taken as 2%, 5%, 10% and 20% of n. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 10 25 50 100 0 1 2 3 4 5 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 200 0 1 2 3 4 5 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 10 25 50 100 0 1 2 3 4 5 6 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 200 0 1 2 3 4 5 6 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 10 25 50 100 0 1 2 3 4 5 6 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 200

0 1 2 3 4 5 6 k
Figure 3: Boxplots of p θ p px 0 q{θ p px 0 q for Model 2 based on 500 replications of sample sizes n " 500 (left column) and n " 1 000 (right column) for p " 1{p2nq. From top to bottom we have x 0 " 0.3, x 0 " 0.5 and x 0 " 0.8. The values of k are taken as 2%, 5%, 10% and 20% of n. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 10 25 50 100 0 1 2 3 4 5 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 200 0 1 2 3 4 5 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 10 25 50 100 0 1 2 3 4 5 6 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 200 0 1 2 3 4 5 6 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 10 25 50 100 0 1 2 3 4 5 6 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 200

0 1 2 3 4 5 6 k
Figure 4: Boxplots of p θ p px 0 q{θ p px 0 q for Model 2 based on 500 replications of sample sizes n " 500 (left column) and n " 1 000 (right column) for p " 1{p5nq. From top to bottom we have x 0 " 0.3, x 0 " 0.5 and x 0 " 0.8. The values of k are taken as 2%, 5%, 10% and 20% of n. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 25 50 100 0 1 2 3 4 5 6 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 200 0 1 2 3 4 5 6 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 25 50 100 0 1 2 3 4 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 200 0 1 2 3 4 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 25 50 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 200 θ p px 0 q{θ p px 0 q for Model 3 based on 500 replications of sample sizes n " 500 (left column) and n " 000 (right column) for p " 1{p2nq. From top to bottom we have x 0 " 3, x 0 " 5 and x 0 " 7. The values of k are taken as 2%, 5%, 10% and 20% of n. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 25 50 100 0 1 2 3 4 5 6 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 200 0 1 2 3 4 5 6 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 25 50 100 0 1 2 3 4 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 200 0 1 2 3 4 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 25 50 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 100 θ p px 0 q{θ p px 0 q for Model 3 based on 500 replications of sample sizes n " 500 (left column) and n " 000 (right column) for p " 1{p5nq. From top to bottom we have x 0 " 3, x 0 " 5 and x 0 " 7. The values of k are taken as 2%, 5%, 10% and 20% of n. economic data of insurance customers along with details about the insured vehicle. This dataset was already used in [START_REF] Girard | Extreme conditional expectile estimation in heavy-tailed heteroscedastic regression models[END_REF] to illustrate the estimation of extreme conditional expectiles for the total claim amount given customer lifetime value and income. A conditional expectile is a risk measure for a single risk factor, which has in the paper by [START_REF] Girard | Extreme conditional expectile estimation in heavy-tailed heteroscedastic regression models[END_REF] been extended to the situation where the risk factor is observed together with a covariate. We estimate the conditional marginal expected shortfall of the total claim amount (i.e., cumulative over the duration of the contract), Y p1q , conditional on the customer lifetime value, Y p2q , exceeding a high quantile and a given value for the covariate income, X. Compared to the analysis in [START_REF] Girard | Extreme conditional expectile estimation in heavy-tailed heteroscedastic regression models[END_REF] we focus on the case of a bivariate risk factor pY p1q , Y p2q q which is observed together with a random covariate, and we estimate the conditional marginal expected shortfall.

In our analysis, we only use the data with a nonzero value for the income variable, leading to n " 6817 observations. The scatterplot of total claim amount versus customer lifetime value, shown in Figure 7, indicates a positive association between these variables. In order to verify the Pareto-type behavior of Y p1q , we construct the local Pareto quantile plots of the Y p1q data for which X P r30000, 40000s and X P r70000, 80000s, respectively, see Figure 8, top row. Clearly, the local Pareto quantile plots become approximately linear in the largest observations, which confirms underlying Pareto-type distributions (see [START_REF] Beirlant | Statistics of Extremes: Theory and Applications[END_REF], for a general discussion of Pareto quantile-quantile plots). Also shown in Figure 8 are the Hill estimates of the total claim amount for which X P r30000, 40000s and X P r70000, 80000s, respectively (bottom row). When focusing on the stable horizontal parts of these plots we can clearly see that the theoretical requirement γ 1 px 0 q ă 0.5 is satisfied. Also for Y p2q the local Pareto quantile plots become linear in the largest observations, indicating underlying Pareto-type distributions, though the linearity is only in the very largest observations; see Figure 9. Similar local Pareto quantile plots were obtained at other incomes.

Next, we investigate the asymptotic dependence assumption by plotting, in Figure 10,p Rp1, 1|x 0 q given in (8) as a function of k, at x 0 " 35000 and x 0 " 75000. In the stable regions of these plots, which are from approximately k " 300 to k " 700, we get confidence intervals for Rp1, 1|x 0 q that do not contain zero, which indicates that Y p1q and Y p2q seem to be asymptotically dependent given X " x 0 . However, this heuristic evaluation cannot be viewed as a rigorous test, as the one proposed, e.g., by [START_REF] Hüsler | Testing asymptotic independence in bivariate extremes[END_REF] in the usual unconditional setting, since, unfortunately, no formal test for asymptotic independence exists in the regression context.

Finally, we illustrate the estimation of the conditional marginal expected shortfall of total claim amount given a customer lifetime value that exceeds a high quantile and given a certain income. In Figure 11, we show p θ p px 0 q as a function of income for p " 0.1% and p " 0.05%. To obtain the estimate for θ p px 0 q, we firstly obtain p γ 1,k 1 px 0 q. This is done by plotting p γ 1,k 1 px 0 q as a function of k 1 , followed by determining k 1 by applying a stability criterion as described in [START_REF] Goegebeur | Bias-corrected estimation for conditional Paretotype distributions with random right censoring[END_REF]. Then, the k-value for the estimation of the conditional marginal expected shortfall at a given x 0 was obtained in a similar way. Note that k and k 1 are determined in an automatic data driven way, as the bounds on α, α 1 , and ∆ in Theorem 2.2 cannot be used in practice, since they depend on unknown parameters. Also the bandwidth parameter h n was determined in a data driven way with the same cross-validation criterion as in the simulation section, resulting in h n " 26950. From Figure 11 we see that smaller values of p lead to larger estimates of the conditional marginal expected shortfall, as expected. Overall, the conditional marginal expected shortfall is quite stable for incomes up to 60000 whereafter it shows a slight decrease. In Figure 11 we show also pointwise 95% confidence intervals for θ p px 0 q based on (7). As expected, the confidence intervals are clearly wider for p " 0.05% than for p " 0.1%, reflecting the higher uncertainty of the estimate at p " 0.05% due to the fact that the estimation is based on fewer observations. Based on the results displayed in Figure 11, one might wonder if the covariate has a significant impact on the marginal expected shortfall. Assessing this formally requires that a process convergence result for p θ p pxq, properly normalised, in terms of x is available, which would then form the basis for the derivation of a test statistic for testing a specific form for θ p pxq. Obtaining such a process result is highly non-trivial. Even in the simpler case of local estimation of the conditional tail index γpxq of a Pareto-type tail with random covariates, it remains uncertain whether this type of result is possible. Our context is much more complicated than this latter univariate framework, and thus the problem is still open.

Preliminary results

To be self contained, we recall below Theorem 2.3 from [START_REF] Goegebeur | Conditional marginal expected shortfall[END_REF] which states the weak convergence of θ k n px 0 q. Note that this theorem has been adjusted due to our Assumption pSq which is slightly different from the one in the latter paper, and the additional Hölder-type Rp1, 1|x 0 q as a function of k, for x 0 " 35000 (left) and x 0 " 75000 (right), with pointwise 95% confidence intervals. Figure 11: Vehicle insurance dataset. p θ p px 0 q along with pointwise 95% confidence intervals for θ p px 0 q at several values of x 0 for p " 0.1% (left), p " 0.05% (right). condition on the R´function in Assumption pHq.

Theorem 5.1 Assume pDq, pHq, pKq, pSq with x Ñ Rpy 1 , y 2 |xq being a continuous function, 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 ´∆ 3 pnq, where 1 and 3 are slowly varying functions at infinity, with α P p0, 1q and

max ˆα d `2γ 1 px 0 qpη R ^ηA 1 ^ηγ 1 q , α d `2p1 ´γ1 px 0 qqpη A 2 ^ηγ 2 ^ηB 2 ^ηε 2 ^ηf X q , α d ´2p1 ´αqγ 2 1 px 0 qβ 1 px 0 q d `dpβ 1 px 0 q `εqγ 1 px 0 q , α ´2p1 ´αqpγ 1 px 0 q ^pβ 2 px 0 qγ 2 px 0 qq ^p´τ qq d ă ∆ ă α d .
Then, for γ 1 px 0 q ă 1{2, we have

b kh d n ˜θ k n px 0 q θ k n px 0 q ´1¸ ´p1 ´γ1 px 0 qq W p8, 1q f X px 0 q `1 f X px 0 q ş 8 0 W ps, 1qds ´γ1 px 0 q ş 8 0 Rps, 1|x 0 qds ´γ1 px 0 q .
22 Now, remark that, assuming that F 1 py|x 0 q is strictly increasing in y, we have

p γ 1,k 1 px 0 q " 1 p f n px 0 q 1 k 1 n ÿ i"1 K hn px 0 ´Xi q ż Y p1q i p U 1 pn{k 1 |x 0 q 1 u du1l tY p1q i ě p U 1 pn{k 1 |x 0 qu " 1 p f n px 0 q ż 8 p U 1 pn{k 1 |x 0 q 1 k 1 n ÿ i"1 K hn px 0 ´Xi q 1 u 1l tY p1q i ěuu du " 1 p f n px 0 q ż 8 p U 1 pn{k 1 |x 0 q 1 k 1 n ÿ i"1 K hn px 0 ´Xi q1l tF 1 pY p1q i |x 0 qď k 1 n n k 1 F 1 pu|x 0 qu 1 u du " γ 1 px 0 q p f n px 0 q ż 1 0 1 k 1 n ÿ i"1 K hn px 0 ´Xi q1l tF 1 pY p1q i |x 0 qď k 1 n n k 1 F 1 pz ´γ1 px 0 q p U 1 pn{k 1 |x 0 q|x 0 qu 1 z dz " γ 1 px 0 q p f n px 0 q ż 1 0 T n pp s n pz|x 0 q|x 0 q 1 z dz, (9)
where

T n py|x 0 q :" 1 k 1 n ÿ i"1 K hn px 0 ´Xi q1l tF 1 pY p1q i |x 0 qďk 1 {n yu ,
and p

s n pz|x 0 q :" n k 1 F 1 ´z´γ 1 px 0 q p U 1 pn{k 1 |x 0 q ˇˇx 0 ¯.
Thus we need to study the asymptotic properties of T n py|x 0 q, p s n pz|x 0 q and p U 1 pn{k 1 |x 0 q.

We start by showing the weak convergence of the process based on T n py|x 0 q, first when the process is centered around its expectation (Theorem 5.2) and then when it is centered around the dominant term of its expectation (Corollary 5.1).

Theorem 5.2 Assume pDq, pHq, pKq, and x 0 P Int(S X q with f X px 0 q ą 0, and y Ñ F 1 py|x 0 q is strictly increasing. Consider sequences k 1 Ñ 8 and h n Ñ 0 as n Ñ 8, in such a way that k 1 {n Ñ 0, k 1 h d n Ñ 8 and h ηγ 1 ^ηε 1 n ln n{k 1 Ñ 0. Then for η P r0, 1{2q, we have,

b k 1 h d n ˆTn py|x 0 q ´EpT n py|x 0 qq y η ˙ W py, 8q y η , (10)
in Dpp0, T sq, for any T ą 0.

Proposition 5.1 Assume pDq, pHq, pKq, and x 0 P Int(S X q with f X px 0 q ą 0. Consider sequences k 1 Ñ 8 and h n Ñ 0 as n Ñ 8, in such a way that k 1 {n Ñ 0, and h ηγ 1 ^ηε 1 n ln n{k 1 Ñ 0. Then for η P r0, 1q, we have,

E pT n py|x 0 qq y η " y 1´η f X px 0 q `O ´hη f X ^ηA 1 n ¯`O ˆhηγ 1 n ln n k 1 ˙`O ˆˇˇˇδ 1 ˆU1 ˆn k 1 ˇˇx 0 ˙ˇˇx 0 ˙ˇˇˇh η B 1 n Ȯ ˆˇˇˇδ 1 ˆU1 ˆn k 1 ˇˇx 0 ˙ˇˇx 0 ˙ˇˇˇh ηε 1 n ln n k 1 ˙,
where the O-terms are uniform in y P p0, T s, for any T ą 0.

Corollary 5.1 Assume pDq, pHq, pKq, x 0 P IntpS X q with f X px 0 q ą 0, and y Ñ F 1 py|x 0 q is strictly increasing. Consider sequences k 1 Ñ 8 and h n Ñ 0 as n Ñ 8, in such a way that k

1 {n Ñ 0, k 1 h d n Ñ 8, h ηε 1 n ln n{k 1 Ñ 0, a k 1 h d n h η f X ^ηA 1 n Ñ 0, a k 1 h d n h ηγ 1 n ln n{k 1 Ñ 0, a k 1 h d n |δ 1 pU 1 pn{k 1 |x 0 q|x 0 q|h η B 1 n
Ñ 0, and

a k 1 h d n |δ 1 pU 1 pn{k 1 |x 0 q|x 0 q|h ηε 1 n ln n{k 1 Ñ 0. Then for η P r0, 1{2q, we have, b k 1 h d n ˆTn py|x 0 q y η ´y1´η f X px 0 q ˙ W py, 8q y η ,
in Dpp0, T sq, for any T ą 0.

In the sequel, for convenient representation, the limiting process in Theorem 5.2 and Corollary 5.1 will be defined on the same probability space as the original random variables, 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, that sup yPp0,T s ˇˇˇb k 1 h d n ˆTn py|x 0 q y η ´y1´η f X px 0 q ˙´W py, 8q y η ˇˇˇÑ 0 a.s. ,

as n Ñ 8.

For the intermediate quantile estimate p U 1 pn{k|x 0 q, we recall Lemma 5.6 from [START_REF] Goegebeur | Conditional marginal expected shortfall[END_REF], which is used several times in our proofs, and which states the weak convergence of p u n :" p U 1 pn{k 1 |x 0 q{U 1 pn{k 1 |x 0 q.

Lemma 5.1 Assume pDq, pHq, pKq, x 0 P IntpS X q with f X px 0 q ą 0 and y Ñ F 1 py|x 0 q is strictly increasing. Consider sequences k 1 Ñ 8 and h n Ñ 0 as n Ñ 8, in such a way that

k 1 {n Ñ 0, k 1 h d n Ñ 8, h ηε 1 n ln n{k 1 Ñ 0, a k 1 h d n h η f X ^ηA 1 n Ñ 0, a k 1 h d n h ηγ 1 n ln n{k 1 Ñ 0, a k 1 h d n |δ 1 pU 1 pn{k 1 |x 0 q|x 0 q| Ñ 0. Then, as n Ñ 8, we have b k 1 h d n pp u n ´1q
γ 1 px 0 qW p1, 8q f X px 0 q .

From Lemma 5.1, we can show now the uniform convergence in probability of p s n pz|x 0 q towards z for any z P p0, T s.

Lemma 5.2 Under the assumptions of Lemma 5.1, for any T ą 0, we have sup zPp0,T s |p s n pz|x 0 q ´z| " o P p1q.

Proof of the preliminary results

Proof of Theorem 5.2. Recall that

T n py|x 0 q y η " 1 k 1 n ÿ i"1 K hn px 0 ´Xi q1l tF 1 pY p1q i |x 0 qď k 1 n yu 1 y η .
The proof of Theorem 5.2 follows the lines of proof of Theorem 2.1 in [START_REF] Goegebeur | Conditional marginal expected shortfall[END_REF]. Below, we only outline the main differences and refer to the latter paper otherwise. To start, we need some notations from empirical process theory with changing function classes, see for instance [START_REF] Van Der Vaart | Weak convergence and empirical processes, with applications to statistics[END_REF]. Let P be the distribution measure of pY p1q , Xq, and denote the expected value under P as P f :" ş f dP for any real-valued measurable function f : R ˆRd Ñ R. For a function class F, let N rs pε, F, L 2 pP qq, denote the minimal number of ε´brackets needed to cover F. The bracketing integral is then defined as J rs pδ, F, L 2 pP qq "

ż δ 0 b ln N rs pε, F, L 2 pP qqdε.
We introduce our sequence of classes F n on R ˆRd as F n :" tpu, zq Ñ f n,y pu, zq, y P p0, T su , where f n,y pu, zq :"

d nh d n k 1 K hn px 0 ´zq1l tF 1 pu|x 0 qď k 1 n yu 1 y η .
Denote also by F n an envelope function of the class F n . Now, according to Theorem 19.28 in van der Vaart (1998), the weak convergence of the stochastic process (10) follows from the following four conditions. Let ρ x 0 be a semimetric, possibly depending on x 0 , making p0, T s totally bounded. We have to prove that sup ρx 0 py,ȳqďδn P pf n,y ´fn,ȳ q 2 ÝÑ 0 for every δ n OE 0, (11)

P F 2 n " Op1q, (12)
P F 2 n 1l tFnąε ? nu ÝÑ 0 for every ε ą 0, (13) and J rs pδ n , F n , L 2 pP qq ÝÑ 0 for every δ n OE 0. ( 14)

We start with verifying condition p11q, with ρ x 0 py, yq :" |y ´y|. Without loss of generality, we may assume that y ď y. We have

P pf n,y ´fn,y q 2 " nh d n k 1 E » -K 2 hn px 0 ´Xq ˜1l tF 1 pY p1q |x 0 qď k 1 n yu y η ´1l tF 1 pY p1q |x 0 qď k 1 n yu y η ¸2fi fl .
We consider now two cases.

Case 1: y ď δ n . We have

˜1l tF 1 pY p1q |x 0 qď k 1 n yu y η ´1l tF 1 pY p1q |x 0 qď k 1 n yu y η ¸2 ď 3 1l tF 1 pY p1q |x 0 qď k 1 n yu y 2η `1l tF 1 pY p1q |x 0 qď k 1 n yu y 2η .
This implies that P pf n,y ´fn,y q 2 ď 3 nh

d n k 1 E ˜K2 hn px 0 ´Xq 1l tF 1 pY p1q |x 0 qď k 1 n yu y 2η ņh d n k 1 E ˜K2 hn px 0 ´Xq 1l tF 1 pY p1q |x 0 qď k 1 n yu y 2η " 3 nh d n k 1 ż R d 1 h 2d n K 2 ˆx0 ´v h n ˙P ´F 1 pY p1q |x 0 q ď k 1 n y|X " v ȳ2η f X pvqdv `nh d n k 1 ż R d 1 h 2d n K 2 ˆx0 ´v h n ˙P ´F 1 pY p1q |x 0 q ď k 1 n y|X " v ȳ2η f X pvqdv " 3 n k 1 ż S K K 2 pvq P ´F 1 pY p1q |x 0 q ď k 1 n y|X " x 0 ´hn v ȳ2η f X px 0 ´hn vqdv `n k 1 ż S K K 2 pvq P ´F 1 pY p1q |x 0 q ď k 1 n y|X " x 0 ´hn v ȳ2η f X px 0 ´hn vqdv. Since P ´F 1 pY p1q |x 0 q ď k 1 n y ˇˇX " x 0 ´hn v ¯" F 1 ´U1 p n k 1 y |x 0 q ˇˇx 0 ´hn v ¯, this yields P pf n,y ´fn,y q 2 ď 3 y 1´2η ż S K K 2 pvqf X px 0 ´hn vqdv `3 ż S K K 2 pvq " 1 y 2η n k 1 F 1 ˆU1 ˆn k 1 y ˇˇx 0 ˙ˇˇx 0 ´hn v ˙´y 1´2η  f X px 0 ´hn vqdv `y1´2η ż S K K 2 pvqf X px 0 ´hn vqdv `żS K K 2 pvq " 1 y 2η n k 1 F 1 ˆU1 ˆn k 1 y ˇˇx 0 ˙ˇˇx 0 ´hn v ˙´y 1´2η  f X px 0 ´hn vqdv.
Using Lemma 5.1 in [START_REF] Goegebeur | Conditional marginal expected shortfall[END_REF] and the fact that ρ x 0 py, yq ď δ n which implies y ď 2δ n , we get

P pf n,y ´fn,y q 2 ď 5 δ 1´2η n ż S K K 2 pvqf X px 0 ´hn vqdv `op1q,
where the op1q´term does not depend on y and y.

Case 2: y ą δ n . In that case

˜1l tF 1 pY p1q |x 0 qď k 1 n yu y η ´1l tF 1 pY p1q |x 0 qď k 1 n yu y η ¸2 ď ˆ1 y η ´1 y η ˙2 1l tF 1 pY p1q |x 0 qď k 1 n yu `1 y 2η ! 1l tF 1 pY p1q |x 0 qď k 1 n yu ´1l tF 1 pY p1q |x 0 qď k 1 n yu
) , from which we deduce that P pf n,y ´fn,y q 2 ď py η ´yη q 2 pyyq 2η ż

S K K 2 pvq n k 1 F 1 pU 1 pn{pk 1 yq|x 0 q|x 0 ´hn vq f X px 0 ´hn vqdv `1 y 2η n k 1 ż S K K 2 pvqP ˆk1 n y ď F 1 pY p1q |x 0 q ď k 1 n y ˇˇˇX " x 0 ´hn v ˙fX px 0 ´hn vqdv.
These two terms on the right-hand side of the above inequality can be handled similarly as those in case 3 in the proof of Theorem 2.1 in [START_REF] Goegebeur | Conditional marginal expected shortfall[END_REF].

Now, a natural envelope function of the class F n is

F n pu, zq :" d nh d n k 1 K hn px 0 ´zq 1l tF 1 pu|x 0 qďk 1 T {nu rpn{k 1 q F 1 pu|x 0 qs η .
Thus, according again to the proof of Theorem 2.1 in [START_REF] Goegebeur | Conditional marginal expected shortfall[END_REF], conditions p12q and p13q are satisfied.

Finally, we need to show condition p14q. Without loss of generality we assume T " 1. Consider for a, θ ă 1: F p1q n paq :" tf n,y P F n : y ď au, and F p2q n p q :" tf n,y P F n : θ `1 ď y ď θ u,

where " 0, . . . , tln a{ ln θu .

The class F p1q n paq has been already studied in the proof of Theorem 2.1 in [START_REF] Goegebeur | Conditional marginal expected shortfall[END_REF] and F p2q n p q can be dealt with similar arguments as for F n p , mq from the latter paper, since we have the following bounds u n pu, zq :"

d nh d n k 1 K hn px 0 ´zq 1l tF 1 pu|x 0 qďk 1 {n θ `1u θ η ď f n,y pu, zq ď d nh d n k 1 K hn px 0 ´zq 1l tF 1 pu|x 0 qďk 1 {n θ u θ p `1qη ": u n pu, zq.
This concludes the proof of Theorem 5.2.

Proof of Proposition 5.1. We have

E rT n py|x 0 qs y η " 1 y η n k 1 E " K hn px 0 ´Xq1l tF 1 pY p1q |x 0 qď k 1 n yu ı " 1 y η n k 1 ż R d 1 h d n K ˆx0 ´u h n ˙P ˆF 1 ´Y p1q ˇˇx 0 ¯ď k 1 n y ˇˇX " u ˙fX puqdu " 1 y η n k 1 ż S K KpuqF 1 ˆU1 ˆn k 1 y ˇˇx 0 ˙ˇˇx 0 ´hn u ˙fX px 0 ´hn uqdu " y 1´η f X px 0 q `y1´η ż S K Kpuq rf X px 0 ´hn uq ´fX px 0 qs du `fX px 0 q ż S K Kpuq » - n k 1 F 1 ´U1 ´n k 1 y ˇˇx 0 ¯ˇˇx 0 ´hn u ȳη ´y1´η fi fl du `żS K Kpuq » - n k 1 F 1 ´U1 ´n k 1 y ˇˇx 0 ¯ˇˇx 0 ´hn u ȳη ´y1´η fi fl rf X px 0 ´hn uq ´fX px 0 qs du.
Following the lines of proof of Lemma 5.1 in [START_REF] Goegebeur | Conditional marginal expected shortfall[END_REF], we have

ˇˇˇˇˇn k 1 F 1 ´U1 ´n k 1 y ˇˇx 0 ¯ˇˇx 0 ´hn u ȳη ´y1´η ˇˇˇˇˇ" O ´hη A 1 n ¯`O ˆhηγ 1 n ln n k 1 ˙`O ˆˇˇˇδ 1 ˆU1 ˆn k 1 ˇˇx 0 ˙ˇˇx 0 ˙ˇˇˇh η B 1 n Ȯ ˆˇˇˇδ 1 ˆU1 ˆn k 1 ˇˇx 0 ˙ˇˇx 0 ˙ˇˇˇh ηε 1 n ln n k 1 ˙,
with O´terms which are uniform in y P p0, T s, for any T ą 0. This yields Proposition 5.1.

Proof of Corollary 5.1. Using the decomposition

b k 1 h d n ˆTn py|x 0 q y η ´y1´η f X px 0 q ˙" b k 1 h d n ˆTn py|x 0 q ´EpT n py|x 0 qq y η ḃk 1 h d n ˆE pT n py|x 0 qq y η ´y1´η f X px 0 q ˙,
combined with Theorem 5.2 and Proposition 5.1 yields Corollary 5.1.

Proofs of the main results

Proof of Theorem 2.1. Using (9), we have the following decomposition b k 1 h d n pp γ 1,k 1 px 0 q ´γ1 px 0 qq "

γ 1 px 0 q f X px 0 q ż 1 0 W pz, 8q 1 z dz `γ1 px 0 q b k 1 h d n ż 1 0 " p s n pz|x 0 q z ´1 dz `γ1 px 0 q f X px 0 q ż 1 0 rW pp s n pz|x 0 q, 8q ´W pz, 8qs 1 z dz `γ1 px 0 q ż 1 0 " b k 1 h d n " T n pp s n pz|x 0 q|x 0 q f X px 0 qp s η n pz|x 0 q ´p s 1´η n pz|x 0 q  ´W pp s n pz|x 0 q, 8q f X px 0 qp s η n pz|x 0 q * p s η n pz|x 0 q z dz ´γ1 px 0 q f X px 0 q p f n px 0 q ż 1 0 T n pp s n pz|x 0 q|x 0 q 1 z dz c k 1 n b nh d n ´p f n px 0 q ´fX px 0 q ": γ 1 px 0 q f X px 0 q ż 1 0 W pz, 8q 1 z dz `4 ÿ i"1 T i,n .
We study each term separately.

Concerning T 1,n , following the lines of proof of Lemma 5.2, we have

T 1,n " γ 1 px 0 q b k 1 h d n " p u ´1{γ 1 px 0 q n ´1ı `oP p1q.
Now, combining Lemma 5.1 with a Taylor expansion, we have T 1,n ´γ1 px 0 q f X px 0 q W p1, 8q.

Concerning T 2,n , for δ P p0, 1q, we use the decomposition T 2,n " γ 1 px 0 q f X px 0 q "ż δ 0 rW pp s n pz|x 0 q, 8q ´W pz, 8qs 1 z dz `ż 1 δ rW pp s n pz|x 0 q, 8q ´W pz, 8qs 1 z dz * ": T |W pp s n pz|x 0 q, 8q ´W pz, 8q| ą ξ ḑ P ˜sup zPrδ,1s

|W pp s n pz|x 0 q, 8q ´W pz, 8q| ą ξ, sup zPrδ,1s

|p s n pz|x 0 q ´z| ď 1 pk 1 h d n q 1{4 P ˜sup zPrδ,1s

|p s n pz|x 0 q ´z| ą 1 pk 1 h d n q 1{4 ḑ P ¨sup zPrδ,1s,|y´z|ď

1 pk 1 h d n q 1{4
|W py, 8q ´W pz, 8q| ą ξ '`P ˜sup zPrδ,1s

|p s n pz|x 0 q ´z| ą 1

pk 1 h d n q 1{4 " op1q,
by the continuity of W p¨, 8q. This implies that

|T p2q 2,n | ď ε ln 1 δ " ´ε η ln ε.
Hence, T 2,n " o P p1q.

Concerning T 3,n , from Lemma 5.2, we have, for n large, with arbitrary large probability |T 3,n | ď γ 1 px 0 q f X px 0 q sup yPp0,2s

ˇˇˇb k 1 h d n ˆTn py|x 0 q y η ´y1´η f X px 0 q ˙´W py, 8q y η ˇˇˇż 1 0 p s η n pz|x 0 q z dz.

Then, by Corollary 5.1 combined with the Skorohod representation theorem, we can conclude that T 3,n " o P p1q.

Finally, T 4,n " o P p1q using the properties of the kernel density estimator.

This achieves the proof of Theorem 2.1.

Proof of Theorem 2.2. We use the decomposition p θ p px 0 q θ p px 0 q " # ˆk np ˙p γ 1,k 1 px 0 q´γ 1 px 0 q

+ loooooooooooooomoooooooooooooon T 5,n # θ k n px 0 q θ k n px 0 q + looooomooooon T 6,n
$ & % p k np q γ 1 px 0 q θ k n px 0 q θ p px 0 q

, .

loooooooooooomoooooooooooon

T 7,n
, from which we deduce that p θ p px 0 q θ p px 0 q ´1 " # ˆk np ˙p γ 1,k 1 px 0 q´γ 1 px 0 q ´1+ T 6,n T 7,n `# θ k n px 0 q θ k n px 0 q ´1+ T 7,n `$ & % p k np q γ 1 px 0 q θ k n px 0 q θ p px 0 q ´1, .

-.( 15)

We will study the three terms pT i,n ´1q, i " 5, 6, 7, separately.

Concerning the term pT 5,n ´1q, remark that, assuming

q ´1+ γ 1 px 0 q f X px 0 q "ż 1 0 W pz, 8q 1 z dz ´W p1, 8q  . (16)
The asymptotic behavior of the term pT 6,n ´1q has been already established in Theorem 5.1. Now, concerning the term pT 7,n ´1q, remark that p k np q γ 1 px 0 q θ k{n px 0 q θ p px 0 q ´1 " ˆθk{n px 0 q{U 1 pn{k|x 0 q θ p px 0 q{U 1 p1{p|x 0 q ´1˙U

1 pn{k|x 0 qp k np q γ 1 px 0 q U 1 p1{p|x 0 q `U1 pn{k|x 0 qp k np q γ 1 px 0 q U 1 p1{p|x 0 q ´1. (17)

Under assumption pDq, (2) yields U 1 pn{k|x 0 qp k np q γ 1 px 0 q U 1 p1{p|x 0 q ´1 " 1 `a1 pn{k|x 0 q 1 `a1 p1{p|x 0 q ´1 " o ˜1 a

kh d n ¸, (18)
since 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 pU 1 p1{p|x 0 q|x 0 q| Ñ 0 under our assumptions. Moreover θ k{n px 0 q U 1 pn{k|x 0 q " ż 8 0 n k P ´Y p1q ą y 1 , Y p2q ě U 2 pn{k|x 0 q ˇˇx 0 ¯dy 1 U 1 pn{k|x 0 q " ż 8 0 n k P ˆ1 ´F1 pY p1q |x 0 q ă 1 ´F1 py 1 |x 0 q, 1 ´F2 pY p2q |x 0 q ď k n ˇˇx 0 ˙dy 1 U 1 pn{k|x 0 q " ż 8 0 R n{k ´n k r1 ´F1 py 1 |x 0 qs , 1 ˇˇx 0 ¯dy 1 U 1 pn{k|x 0 q " ´ż 8 0 R n{k ´n k " 1 ´F1 pz ´γ1 px 0 q 1 U 1 pn{k|x 0 q|x 0 q ı , 1 ˇˇx 0 ¯dz ´γ1 px 0 q 1 " ´ż 8 0 Rpz 1 , 1|x 0 qdz ´γ1 px 0 q 1 ´ż 8 0 " R n{k ´n k " 1 ´F1 pz ´γ1 px 0 q 1 U 1 pn{k|x 0 q|x 0 q ı , 1 ˇˇx 0 R ´n k " 1 ´F1 pz ´γ1 px 0 q 1 U 1 pn{k|x 0 q|x 0 q ı , 1 ˇˇx 0 ¯ı dz ´γ1 px 0 q 1 ´ż 8 0 " R ´n k " 1 ´F1 pz ´γ1 px 0 q 1 U 1 pn{k|x 0 q|x 0 q ı , 1 ˇˇx 0 ¯´R pz 1 , 1|x 0 q ı dz ´γ1 px 0 q 1 ": ´ż 8 0 Rpz 1 , 1|x 0 qdz ´γ1 px 0 q 1 `T8,n `T9,n .

we have θ p px 0 q U 1 p1{p|x 0 q " ´ż 8 0 Rpz 1 , 1|x 0 qdz ´γ1 px 0 q 1 `o ˜1 a kh d n ¸.

(20)

Combining ( 17), ( 18), ( 19) and ( 20), we deduce that p k np q γ 1 px 0 q θ k{n px 0 q θ p px 0 q ´1 " o ˜1 a kh d n ¸.

Finally, decomposition (15) combined with Theorem 5.1, ( 16) and ( 21) achieves the proof of Theorem 2.2.

Figure 2 :

 2 Figure2: Boxplots of p θ p px 0 q{θ p px 0 q for Model 1 based on 500 replications of sample sizes n " 500 (left column) and n " 1 000 (right column) for p " 1{p5nq. From top to bottom we have x 0 " 3, x 0 " 5 and x 0 " 7. The values of k are taken as 2%, 5%, 10% and 20% of n.

Figure 5 :

 5 Figure5: Boxplots of p θ p px 0 q{θ p px 0 q for Model 3 based on 500 replications of sample sizes n " 500 (left column) and n " 000 (right column) for p " 1{p2nq. From top to bottom we have x 0 " 3, x 0 " 5 and x 0 " 7. The values of k are taken as 2%, 5%, 10% and 20% of n.

Figure 6 :

 6 Figure6: Boxplots of p θ p px 0 q{θ p px 0 q for Model 3 based on 500 replications of sample sizes n " 500 (left column) and n " 000 (right column) for p " 1{p5nq. From top to bottom we have x 0 " 3, x 0 " 5 and x 0 " 7. The values of k are taken as 2%, 5%, 10% and 20% of n.

Figure 7 :

 7 Figure 7: Vehicle insurance dataset. Scatterplot of total claim amount versus customer lifetime value.

Figure 8 :Figure 9 :Figure 10 :

 8910 Figure8: Vehicle insurance dataset. Top row: Pareto quantile plots of the total claim amount for which income P r30000, 40000s (left) and income P r70000, 80000s (right). Bottom row: Hill estimates of the total claim amount for which income P r30000, 40000s (left) and income P r70000, 80000s (right).

δ

  with Potter's bounds (see Proposition B.1.9 (5) in de[START_REF] De Haan | Extreme value theory, an introduction[END_REF], we have for any ζ P p0, 1{γ 1 px 0 qq, for n large, and with arbitrary large probability p1´γ 1 px 0 qζqη `Cδ η " Cε 1´γ 1 px 0 qζ `Cε, by choosing δ η " ε. Now, concerning T p2q 2,n , remark that following the lines of proof of Lemma 5.2pz|x 0 q ´z| " o P p1q, from which we deduce that, for any ξ ą 0 P ˜sup zPrδ,1s

  .

	x 0	p " 1{n	p " 1{p2nq	p " 1{p5nq
	Model 1 3 0.946 (0.954) 0.946 (0.954) 0.948(0.954)
	5 0.956 (0.960) 0.960 (0.960) 0.962 (0.958)
	7 0.958 (0.936) 0.956 (0.936) 0.956 (0.936)
	Model 2 0.3 0.842 (0.906) 0.842 (0.906) 0.842 (0.906)
	0.5 0.838 (0.954) 0.840 (0.952) 0.844 (0.954)
	0.8 0.956 (0.936) 0.958 (0.936) 0.958 (0.934)
	Model 3 3 0.938 (0.962) 0.942 (0.964) 0.940 (0.964)
	5 0.880 (0.956) 0.880 (0.958) 0.876 (0.958)
	7 0.854 (0.954) 0.856 (0.948) 0.854 (0.944)
	Table

  ln k{pnpq ? px 0 q´γ 1 px 0

									n k 1 h d	ÝÑ 0 and using Theorem 2.1,
	we have
	ˆk np	˙p γ 1,k 1 px 0 q´γ 1 px 0 q	´1 " exp	# b	k 1 h d n rp γ 1,k 1 px 0 q ´γ1 px 0 qs	ln k{pnpq a n k 1 h d	+	´1
						"	b k 1 h d n rp γ 1,k 1 px 0 q ´γ1 px 0 qs	ln k{pnpq a n k 1 h d	p1 `oP p1qq ,
	from which we deduce that
	a k 1 h d n ln k{pnpq	#	ˆk np	˙p γ 1,k 1

Acknowledgement

The authors would like to thank the referees, the Associate Editor and editor for their helpful comments. 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).

Proof of Lemma 5.2. We have, for z P p0, T s and any ε ą 0 and ζ P p0, β 1 px 0 qs |p s n pz|x 0 q ´z| " ˇˇˇn k 1 F 1 ´z´γ 1 px 0 q p u n U 1 pn{k 1 |x 0 q ˇˇx 0 ¯´z ˇˇ" ˇˇˇˇˇF 1 ´z´γ 1 px 0 q p u n U 1 pn{k 1 |x 0 q ˇˇx 0 F 1 ´U1 pn{k 1 |x 0 q ˇˇx 0 ¯´z ˇˇˇˇď

for n large, with arbitrary large probability, by Proposition B.1.10 in de [START_REF] De Haan | Extreme value theory, an introduction[END_REF]. In the above, the notation a ˘' means a ' if a ě 1 and a ´' if a ă 1. Using Lemma 5.1, Lemma 5.2 follows.

A similar type of property can be obtained for θ p px 0 q instead of θ k{n px 0 q. Indeed

Clearly T 10,n " Opp ´τ q " o ˆ1 ?

for n large. Then, if α and ∆ are chosen as stated in Theorem 2.2 and T n " n κ with κ chosen such that α ´∆d 2γ 1 px 0 q ă κ ă 2p1 ´αqγ 1 px 0 qβ 1 px 0 q ´pα ´∆dq 2r1 ´γ1 px 0 q `pβ 1 px 0 q `εqγ 1 px 0 qs , 45.2449&rep=rep1&type=pdf.