A Usseglio-Carleve

ESTIMATION OF CONDITIONAL EXTREME RISK MEASURES FROM HEAVY-TAILED ELLIPTICAL RANDOM VECTORS

Keywords: Elliptical distribution, Extreme quantiles, Extreme value theory, Haezendonck-Goovaerts risk measures, Heavy-tailed distributions, L p -quantiles

In this work, we focus on some conditional extreme risk measures estimation for elliptical random vectors. In a previous paper, we proposed a methodology to approximate extreme quantiles, based on two extremal parameters. We thus propose some estimators for these parameters, and study their consistency and asymptotic normality in the case of heavy-tailed distributions. Thereafter, from these parameters, we construct extreme conditional quantiles estimators, and give some conditions that ensure consistency and asymptotic normality. Using recent results on the asymptotic relationship between quantiles and other risk measures, we deduce estimators for extreme conditional Lp-quantiles and Haezendonck-Goovaerts risk measures. Under similar conditions, consistency and asymptotic normality are provided. In order to test the effectiveness of our estimators, we propose a simulation study. A financial data example is also proposed.

Introduction

In many fields such as finance or actuarial science, quantile, or Value-at-Risk (see [START_REF] Linsmeier | Value at risk[END_REF]) is a recognized tool for risk measurement. In [START_REF] Koenker | Regression quantiles[END_REF], quantile is seen as minimum of an asymmetric loss function. However, Value-at-Risk, or VaR, has some disadvantages, such as that of not being a coherent measure in the sense of [START_REF] Artzner | Coherent measures of risk[END_REF]. These limits have led many authors to use alternative risk measures. On the basis of Koenker's approach, [START_REF] Newey | Asymmetric least squares estimation and testing[END_REF] proposed another measure called expectile, which has since been widely studied (see for example [START_REF] Sobotka | Geoadditive expectile regression[END_REF] or more recently Daouia et al. (2017a)) and applied [START_REF] Taylor | Estimating Value at Risk and Expected Shortfall Using Expectiles[END_REF] and [START_REF] Cai | Optimal reinsurance with expectile[END_REF]). Later, [START_REF] Breckling | M-quantiles[END_REF] introduced M-quantiles, a family of measures minimizing an asymmetric loss function, and [START_REF] Chen | Conditional Lp-quantiles and their application to the testing of symmetry in nonparametric regression[END_REF] focused on asymmetric power functions to define L p -quantiles. The cases p = 1 and p = 2 correspond respectively to the quantile and expectile. Recently, [START_REF] Bernardi | On the Lp-quantiles for the Student t distribution[END_REF] provided some results concerning L p -quantiles for Student distributions, and have shown that closed formula are difficult to obtain in the general case. In parallel, [START_REF] Artzner | Coherent measures of risk[END_REF] introduced the Tail-Value-at-Risk as an alternative to Value-at-Risk, and this risk measure subsequently had many applications (see e.g. [START_REF] Bargès | TVaR-based capital allocation with copulas[END_REF]). Moreover, TVaR belongs to a larger family of risk measures called Haezendonck-Goovaerts risk measures and introduced in [START_REF] Haezendonck | A new premium calculation principle based on Orlicz norms[END_REF], [START_REF] Goovaerts | Some new classes of consistent risk measures[END_REF] and [START_REF] Tang | On the Haezendonck-Goovaerts risk measure for extreme risks[END_REF]. In the same way as L p -quantiles, we do not have an explicit formula in the general case. However, for a heavy-tailed random variable, [START_REF] Daouia | Extreme M-quantiles as risk measures: from L1 to Lp optimization[END_REF] proved that L p -quantile and L 1 -quantile (or quantile) are asymptotically proportional. Then, as proposed in Daouia et al. (2017a), an estimator of a L p -quantile may be deduced from a suitable estimator of the quantile, for extreme levels. In the same spirit, [START_REF] Tang | On the Haezendonck-Goovaerts risk measure for extreme risks[END_REF] provided a similar asymptotic relationship between a subclass of Haezendonck-Goovaerts risk measures and quantiles. Finally, all these risk measures we introduced may be estimated through a quantile estimation in an asymptotic setting. Extreme quantiles estimation is a very active area of research. In recent years, we can give many examples : [START_REF] Gardes | Estimating extreme quantiles of weibull tail distributions[END_REF] focused on Weibull tail distributions, El [START_REF] El Methni | Estimation of extreme quantiles from heavy and light tailed distributions[END_REF] proposed a study for heavy and light tailed distributions, [START_REF] Gong | Estimation of extreme quantiles for functions of dependent random variables[END_REF] was interested in functions of dependent variables, and de Valk (2016) provided a methodology for high quantiles estimation. The question of extreme conditional quantiles estimation has also been explored in [START_REF] Wang | Estimation of high conditional quantiles for heavy-tailed distributions[END_REF] in a regression framework. However, [START_REF] Maume-Deschamps | Quantile predictions for elliptical random fields[END_REF] and [START_REF] Maume-Deschamps | Spatial expectile predictions for elliptical random fields[END_REF] have shown that the regression setting may lead to a poor estimation of extreme measures in the case of elliptical distributions. Elliptical distributions, introduced in [START_REF] Kelker | Distribution theory of spherical distributions and a location-scale parameter generalization[END_REF], aim to generalize the gaussian distribution, i.e to define symmetric distributions with different properties, such as a heavy tail. This is why elliptical distributions are more and more used in finance (see for example [START_REF] Owen | On the class of elliptical distributions and their applications to the theory of portfolio choice[END_REF] or [START_REF] Xiao | A Black-Litterman asset allocation under Elliptical distributions[END_REF]). For all these reasons, we consider, in this paper, an elliptical random vector Z = (X, Y ) with the consistency property (in the sense of [START_REF] Kano | Consistency property of elliptical probability density functions[END_REF], where X ∈ R N , Y ∈ R, and propose to estimate some extreme quantiles (and deduce L p -quantiles and Haezendonck-Goovaerts risk measures) of Y |X = x, i.e. of a component conditionnally to the others. In order to improve the conditional quantile estimation, we proposed in [START_REF] Maume-Deschamps | Quantile predictions for elliptical random fields[END_REF] a methodology based on two extremal parameters, and the unconditional quantile of Y . Indeed, if we denote F -1 Y |x (α) the quantile of level α of Y |X = x, the latter is asymptotically equivalent to a quantile of Y (F -1 Y will be the quantile function of Y ), in the following manner :

(1.1)

F -1 Y |x (α) ∼ α→1 F -1 Y (δ(α, η, )) ,
where δ is a known function (detailed later) depending on α and two parameters η and called extremal parameters. One can notice that Equation (1.1) may only holds under the consistency property of Z. [START_REF] Maume-Deschamps | Quantile predictions for elliptical random fields[END_REF] has also shown that extremal parameters do not exist for some consistent elliptical distributions (see e.g. the Laplace distribution).

In this paper, the goal will be in a first time to give a sufficient condition on Z that ensures the existence of η and . This is why a regularly varying assumption is done. After having proved their existence, estimators for the parameters η and are proposed, and therefore for extreme conditional quantiles.

The paper is organized as follows. Section 2 provides some definitions and properties of elliptical distributions, including the extremal parameters introduced in Maume-Deschamps et al. (2017). A particular interest is given to consistent elliptical distributions. Section 3 is devoted to extremal parameters η and . Under a regularly varying assumption, their existence is proved, and estimators are proposed. By adding some conditions, consistency and asymptotic normality results are given. In Section 4, we use the results of Section 3 to introduce some estimators of extreme quantiles, and give consistency and asymptotic normality results. The asymptotic relationships between L p -quantiles and quantiles recalled in Section 5 allow us to give extreme L p -quantiles estimators. The same approach is proposed for extreme Haezendonck-Goovaerts risk measures. In order to analyze the efficiency of our estimators, we propose a simulation study in Section 6, and a real data example in Section 7.

Preliminaries

In this section, we first recall some classical results on elliptical distributions. We consider a d-dimensional vector Z from an elliptical distribution with parameters µ ∈ R d and Σ ∈ R d×d . Then the density of Z, if it exists, is given by :

(2.1)

c d |Σ| 1 2 g d (z -µ) T Σ -1 (z -µ) .
c d and g d will respectively be called normalization coefficient and generator of Z. [START_REF] Cambanis | On the theory of elliptically contoured distributions[END_REF] gives another way to characterize an elliptical distribution, through the following stochastic representation :

(2.2) d) ,

Z d = µ + RΛU (
where d) is a d-dimensional random vector uniformly distributed on the unit sphere of dimension d, and R is a non-negative random variable independent of U (d) . R is called radius of Z. In the following, the radius must have a particular shape. Indeed, [START_REF] Huang | Spherically invariant processes: Their nonlinear structure, discrimination, and estimation[END_REF] and [START_REF] Kano | Consistency property of elliptical probability density functions[END_REF] propose a representation for some particular elliptical distributions. Let us consider (Z d ) d∈N * a family of elliptical distributions of dimension d. Then (Z d ) d∈N * possesses the consistency property if it admits the following representation for all d ∈ N * :

ΛΛ T = Σ, U (
(2.3)

Z d d = µ + χ d ξΛU (d) ,
where χ d is the square root of a χ 2 distribution with d degrees of freedom, ξ is a non-negative random variable which does not depend on d, and χ d , ξ and U (d) are mutually independent. In [START_REF] Kano | Consistency property of elliptical probability density functions[END_REF], such elliptical distributions are said consistent, have the advantage of being stable by linear combinations (combining Theorem 2.16 of [START_REF] Fang | Symmetric multivariate and related distributions[END_REF] and Theorem 1 in [START_REF] Kano | Consistency property of elliptical probability density functions[END_REF]), and allow us to define elliptical random fields (see, e.g., [START_REF] Opitz | Modeling asymptotically independent spatial extremes based on Laplace random fields[END_REF]). In the following, we focus on consistent elliptical distributions, and take the notation

(2.4) R d = χ d ξ.
For the sake of clarity, we will say that a random variable with stochastic representation (2.3) is (ξ, d)-elliptical with parameters µ and Σ. Using this terminology, the purpose of the paper is as follows.

Let Z = (X, Y ) ∈ R N +1 be a (ξ, N + 1)-elliptical random vector with parameters µ and Σ, where X ∈ R N and Y ∈ R. Consistency property of Z implies that X and Y are respectively (ξ, N )-and (ξ, 1)-elliptical distributions with parameters µ

X ∈ R N , Σ X ∈ R N ×N and µ Y ∈ R, Σ Y ∈ R.
We also denote Σ XY the covariance vector between X and Y . The aim is thus to provide a predictor for the quantile of the conditional distribution Y |X = x. According to Theorem 7 of Frahm ( 2004), such a distribution is still elliptical, with a radius R * different from R in the general case. In particular, we have :

(2.5) 1) ,

{Y |X = x} d = µ Y |X + σ Y |X R * U (
where µ Y |X = µ Y + Σ T XY Σ X -1 (x -µ X ) and σ 2 Y |X = Σ Y -Σ T XY Σ X -1 Σ XY . Then, denoting Φ R * (t) = P R * U (1)
≤ t , and using the translation equivariance and positive homogeneity of elliptical quantiles (see [START_REF] Mcneil | Quantitative risk management : Concepts, techniques and tools[END_REF]), conditional quantiles of Y |X = x may be expressed as :

(2.6)

q α (Y |X = x) = µ Y |X + σ Y |X Φ -1 R * (α)
, where α ∈]0, 1[. Thus, in order to give a good prediction of q α (Y |X = x), we need to estimate the conditional function Φ -1 R * . Unfortunately, when we have a data set X 1 , ..., X n , we only observe the unconditional distribution of X. This is why, in [START_REF] Maume-Deschamps | Quantile predictions for elliptical random fields[END_REF], we have given a predictor for conditional quantiles, based solely on the unconditional c.d.f Φ R (t) = P R 1 U (1) ≤ t . This approximation is based on two parameters η ∈ R and 0 < < +∞ such that :

(2.7) lim t→∞ ΦR * (t) ΦR (t η ) = .
Table 1 gives some examples of coefficients η and for classical elliptical distributions. However, we have

Distribution η Gaussian 1 1 Student, ν > 0 N ν + 1 Γ( ν+N +1 2 )Γ( ν 2 ) Γ( ν+N 2 )Γ( ν+1 2 ) 1 + M (x) ν N +ν 2 ν N 2 +1 ν+N UGM 1 min(θ 1 ,...,θn) N exp - min(θ 1 ,...,θn) 2 2 M(x) n k=1 π k θ N k exp - θ 2 k 2 M (x) Slash, a > 0 N a + 1 Γ( N +1+a 2 )M(x) N +a 2 Γ( N +a 2 )(N+a)χ 2 N +a (M (x))2 a 2 -1 Γ( 1+a
2 ) Table 1. Coefficients η and for classical distributions, where

M (x) = (x - µ X ) Σ X -1 (x -µ X ).
shown in [START_REF] Maume-Deschamps | Quantile predictions for elliptical random fields[END_REF] that such parameters not always exist for all elliptical distribution (see, e.g, Laplace distribution). In a first time, we can wonder in which setting these parameters exist. We thus consider the following assumption, that will ensures the existence of η and .

Assumption 1 (Second order regular variations). We assume that there exist a function A such that A(t) → 0 as t → +∞, and

(2.8) lim t→+∞ Φ -1 R (1-1 ωt ) Φ -1 R (1-1 t ) -ω γ A(t) = ω γ ω ρ -1 ρ ,
where γ > 0 and ρ < 0.

This assumption is widespread in literature of extreme quantiles (see, e.g, Daouia et al. (2017a)). A first consequence is that Φ R , or equivalently F R1 is attracted to the maximum domain of Pareto-type distributions with tail index γ. Furthermore, it entails Φ -1 R (1-1/t) ∼ c 1 t γ , or equivalently ΦR (t) ∼ c 2 t -1 γ as t → +∞ (see de [START_REF] De Haan | Extreme value theory: an introduction[END_REF]). As example, Student distribution satisfies Assumption 1.

The following lemma provides some results concerning asymptotic equivalences.

Lemma 2.1 (Regular variation properties). Under Assumption 1, we get the following regular variations properties :

(i) The random variable ξ satisfies

(2.9) Fξ (t) ∼ t→+∞ λt -1 γ , λ ∈ R.
(ii) For all d ∈ N * , the random variable R d = χ d ξ is attracted to the maximum domain of Pareto-type distribution with tail index γ, and

(2.10) FR d (t) ∼ t→+∞ 2 1 2γ Γ d+γ -1 2 Γ d 2 Fξ (t) ∼ t→+∞ 2 1 2γ Γ d+γ -1 2 Γ d 2 λt -1 γ , λ ∈ R. (iii) For all η > 0, d ∈ N * , (2.11) f R d (t) f R1 (t η ) ∼ t→+∞ √ πΓ d+γ -1 2 Γ d 2 Γ 1+γ -1 2 t (η-1)(γ -1 +1) .
These results will be usefull throughout the paper, and especially in the following result which proves the existence of our parameters.

Proposition 2.2 (Existence of extremal parameters). Under Assumption 1, parameters η and exist, and are expressed :

(2.12)

   η = 1 + γN = Γ N +γ -1 +1 2 Γ γ -1 +1 2 γ -1 π -N 2 (N +γ -1 )c N g N (M (x))
.

One can notice that η is only related to the tail index γ, and not to the covariate vector x, while is depending on c N g N (M (x)). In the next, we thus denote rather (x), in order to emphasize the role played by the covariate vector x. We can now give the following predictor for q α (Y |X = x) :

(2.13)

q α↑ (Y |X = x) = µ Y |X + σ Y |X Φ -1 R 1 - 1 (x) 1-α + 2(1 -(x)) 1/η .
From there, we have proved in Theorem 7 of [START_REF] Maume-Deschamps | Quantile predictions for elliptical random fields[END_REF] that q α↑ (Y |X = x) and q α (Y |X = x) were asymptoticaly equivalent as α → 1, i.e

(2.14)

q α↑ (Y |X = x) ∼ α→1 q α (Y |X = x).
A similar equivalence has been easily deduced for α → 0, using the symmetry properties of elliptical distributions. In this paper, we focus on the case α → 1, case α → 0 being easily deduced. In Section 3, we propose some estimators for extremal parameters η and (x). Before that, we need to do a little simplification. Indeed, Equation (2.13) shows that the extreme quantile estimation requires the prior estimation of quantities µ Y |X and σ Y |X . These quantities may be easily estimated by the method of moments or fixed-point algorithm (c.f p.66 of Frahm ( 2004)). In a spatial setting, even if the variable Y is not observed, a stationarity assumption on the random field makes it possible to estimate these values (see [START_REF] Cressie | Spatial prediction and ordinary kriging[END_REF]). Furthermore, the speed of convergence of these methods is higher than those of the estimators we propose in this paper, and therefore do not interfere in the asymptotic results. This is why, in the following, we suppose that µ Y |X , σ Y |X , and therefore µ X , Σ X are known. Then, it remains to estimate η, (x) and Φ -1 R * . Section 3 focuses on η and (x), while Section 4 deals with Φ -1 R * .

Extremal coefficients estimation

In this section, the aim is to estimate the extremal parameters η and (x) conditionaly to the covariates vector X = x. For that purpose, we consider a random sample X 1 , ..., X n independent and identically distributed from an (ξ, N )-elliptical vector with the same distribution as X, and denote

M (x) = (x -µ X ) T Σ X -1 (x -µ X ).
The aim is then to give two suitable estimator η and ˆ (x), respectively for η and (x).

3.1. Estimation of η. We notice that coefficient η is directly related on the tail index γ. Then, using a suitable estimator of γ, we easily deduce η. There are several estimators widespread in the literature. As examples, [START_REF] Pickands | Statistical inference using extreme order statistics[END_REF], [START_REF] Schultze | On least squares estimates of an exponential tail coefficient[END_REF] or [START_REF] Kratz | The qq-estimator and heavy tails[END_REF] provide some estimators for γ. In the following, we use the Hill estimator, introduced in Hill (1975) :

(3.1) γkn = 1 k n kn i=1 ln W [i] W [kn+1] ,
where

W [1] ≥ . . . ≥ W [kn+1] ≥ . . . ≥ W [n] and k n = o(n) such that k n → +∞ as n → +∞.
In this context, the statistic W may be :

• The first (or indifferently any) component of the reduced centered covariate vector Λ

X -1 (X -µ X ),
where Λ X T Λ X = Σ X . This approach works well, but we do not use all available data.

• The Mahalanobis norm (X -µ X ) T Σ X -1 (X -µ X ). This approach has the advantage of using all available data. Indeed, according to Theorem 2 of [START_REF] Hashorva | Sample extremes of l p -norm asymptotically spherical distributions[END_REF], the two last quantities both admit γ as tail index.

In the following we will use the one-component approach, since the asymptotic results we give are valid under Assumption 1, applied to the univariate c.d.f Φ R . Moreover, numerical comparisons seem show that the second approach does not significantly improve the estimation of the parameters. Main properties of γkn may be found in de [START_REF] De Haan | On asymptotic normality of the hill estimator[END_REF]. Under second order condition given in Assumption 1, de [START_REF] De Haan | Extreme value theory: an introduction[END_REF] proved the following asymptotic normality for γkn .

(3.2) k n (γ kn -γ) → n→+∞ N λ 1 -ρ , γ 2 , where λ = lim n→+∞ √ k n A n kn and k n = o(n) such that k n → +∞ as n → +∞.
Then, using Proposition 2.2 and Equation (3.1), we define the following estimator for η. Definition 3.1 (Estimator of η). We define ηkn as

(3.3) ηkn = N k n kn i=1 ln W [i] W [kn+1] + 1.
As an affine transformation of Hill estimator, asymptotic normality of ηkn is obvious. In order to simplify the next results, we suppose λ = 0 in what follows.

Proposition 3.1 (Asymptotic normality of ηkn ). Under Assumption 1, and if

lim n→+∞ √ k n A n kn = 0, then (3.4) k n (η kn -η) → n→+∞ N 0, N 2 γ 2 .
3.2. Estimation of (x). The form of (x), given in Proposition 2.2, leads to a more complicated estimation. Indeed, (x) is related on both γ and c N g N (M (x)). Our estimator for γ is given in Equation (3.1). Concerning c N g N (M (x)), we propose a kernel estimator. Class of kernel estimators, introduced in [START_REF] Parzen | On estimation of a probability density function and mode[END_REF], makes it possible to estimate probability densities. Then, the following lemma will be usefull for the construction of our estimator. This result comes from p.108 of [START_REF] Johnson | Multivariate Statistical Simulation[END_REF].

Lemma 3.2. The Mahalanobis distance M (X) = (X -µ X ) T Σ X -1 (X -µ X ) has density : (3.5) f M (X) (t) = π N 2 Γ N 2 x N 2 -1 c N g N (t).
Using Lemma 3.2, we introduced a kernel estimator ĝhn for c N g N (M (x)).

Definition 3.2 (Generator estimator). We define ĝhn as

(3.6) ĝhn = M (x) 1-N 2 Γ N 2 π N 2 fM(X) (M (x)) = M (x) 1-N 2 Γ N 2 π N 2 nh n n i=1 K M (x) -(X i -µ X ) T Σ X -1 (X i -µ X ) h n ,
where the kernel K fills some conditions given in [START_REF] Parzen | On estimation of a probability density function and mode[END_REF] and bandwith h n verifies h n → 0 and nh n → +∞ as n → +∞. [START_REF] Parzen | On estimation of a probability density function and mode[END_REF] provided the asymptotic normality for kernel estimators. We first define some assumptions concerning K and g N needed for the next results.

• (K1) : K is compactly supported on [-1, 1] and bounded. In addition,

R K(u)du = 1, K(u) = K(-u) ∀u ∈ R and R u 2 K(u)du = 0. • (K2) :
In the neighborhood of M (x), g N is bounded and twice continuously differentiable with bounded derivatives. The following results may be found in [START_REF] Li | Nonparametric econometrics: theory and practice[END_REF]. Under conditions (K1) -(K2), it may be proved that :

(3.7)    E fM(X) (M (x)) -f M (X) (M (x)) = O(h 2 n ) Var fM(X) (M (x)) = O 1 nhn .
By adding the condition nh 5 n → 0 as n → +∞, we also obtain the asymptotic normality :

(3.8) nh n fM(X) (M (x)) -f M (X) (M (x)) → n→+∞ N 0, f M (X) (M (x)) K(u) 2 du .
Using the previous results given above, the following asymptotic normality for ĝhn is easily deduced.

Proposition 3.3 (Asymptotic normality of generator estimator). Under conditions (K1) -(K2), and taking a sequence h n such that h n → 0, nh n → +∞ and nh 5 n → 0 as n → +∞, then the following relationship holds :

(3.9) nh n (ĝ hn -c N g N (M (x))) → n→+∞ N 0, M (x) 1-N 2 Γ N 2 π N 2 c N g N (M (x)) K(u) 2 du .
Replacing γ by γ and c N g N (M (x)) by ĝhn in Equation (2.12), we are now able to provide an estimator ˆ (x) for (x), in the following definition. Furthermore, under Assumption 1, we give the asymptotic normality of ˆ (x).

Definition 3.3 (Estimator of (x)). We define ˆ kn,hn (x) as :

(3.10) ˆ kn,hn (x) = Γ N +γ -1 kn +1 2 Γ γ-1 kn +1 2 γ-1 kn π -N 2 N + γ-1 kn ĝhn
.

where γkn and ĝhn are respectively given in Equations (3.1) and (3.6).

Proposition 3.4. Under Assumption 1, conditions (K1) -(K2) and if

lim n→+∞ √ k n A n kn = 0, the following asymptotic relationships hold : (i) If nh n /k n → n→+∞ +∞ and √ k n h 2 n → n→+∞ 0, then (3.11) k n ˆ kn,hn (x) -(x) → n→+∞ N (0, V 1 (γ, c N g N (M (x)))) . (ii) If nh n /k n → n→+∞ 0 and nh 5 n → n→+∞ 0, then (3.12) nh n ˆ kn,hn (x) -(x) → n→+∞ N (0, V 2 (γ, c N g N (M (x)))) ,
where (Ψ is the digamma function (see p.258 of [START_REF] Abramowitz | Handbook of mathematical functions[END_REF]))

(3.13)            V 1 (γ, c N g N (M (x))) = π -N γ 2 c 2 N g N (M (x)) 2 Γ N +γ -1 +1 2 2 Γ γ -1 +1 2 2 Ψ γ -1 +1 2 -Ψ N +γ -1 +1 2 2γ 2 (N γ+1) - N (N γ+1) 2 2 V 2 (γ, c N g N (M (x))) = Γ( N 2 ) M (x) N 2 -1 π N 2 c N g N (M (x)) K(u) 2 du Γ N +γ -1 +1 2 Γ γ -1 +1 2 γ -1 π -N 2 (N +γ -1 )c 2 N g N (M (x)) 2 2 .
We have the asymptotic normality for our estimators η kn and ˆ kn,hn (x). The next proposition gives the joint distribution according to the asymptotic relations between k n and h n . The proof derives from delta method. Proposition 3.5. Under Assumption 1, conditions (K1) -(K2) and if √ k n A n kn → 0 as n → +∞, then the following asymptotic relationships hold :

(i) If nh n /k n → n→+∞ 0 and nh 5 n → n→+∞ 0, then (3.14) nh n ˆ kn,hn (x) -(x) ηkn -η → n→+∞ N 0 0 , V 2 (γ, c N g N (M (x))) 0 0 0 , where V 2 (γ, c N g N (M (x))) is given in Equation (3.13). (ii) If nh n /k n → n→+∞ +∞ and √ k n h 2 n → n→+∞ 0, then (3.15) k n ˆ kn,hn (x) -(x) ηkn -η → n→+∞ N 0 0 , V 1 (γ, c N g N (M (x))) -N γ V 1 (γ, c N g N (M (x))) -N γ V 1 (γ, c N g N (M (x))) N 2 γ 2 , where V 1 (γ, c N g N (M (x))) is given in Equation (3.13).
Using the previous results, we propose, in Section 4, some estimators of extreme conditional quantiles based on ˆ kn,hn (x) and ηkn .

Extreme quantiles estimation

In this section, we propose some estimators of extreme quantiles q αn (Y |X = x), for a sequence α n → 1 as n → +∞. For that purpose, we divide the study in two cases :

• Intermediate quantiles, i.e we suppose n(1 -α n ) → +∞. It entails that the estimation of the α n -quantile leads to an interpolation of sample results. • High quantiles. According to de Haan and Rootzén (1993), we suppose n(1 -α n ) → 0, i.e we need to extrapolate sample results to areas where no data are observed. In both cases, the asymptotic results require some conditions we will provide throughout the section. The first one brings together the assumptions of Proposition 3.5.

• (C) : Kernel conditions (K1) -(K2) hold. In addition, k n → +∞, h n → 0, k n = o(nh n ), √ k n h 2 n → 0 and √ k n A n kn → 0 as n → +∞.
Condition (C) will be common to both approaches, and ensures in a first time that Hill estimator is unbiased, according to Equation (3.2). Moreover, k n = o(nh n ) means that ĝhn converges to c N g N (M (x)) faster than γkn to γ. In practice, this condition seems appropriate, because k n must not be too large for the Hill estimator to be unbiased, and h n must be tall enough to provide a good estimation of (x).

4.1. Intermediate quantiles. We consider the case where n(1

-α n ) → +∞ with α n → 1 as n → +∞. We recall q αn (Y |X = x) = µ Y |X + σ Y |X Φ -1 R * (α n ). According to Equation (2.14), we can approximate Φ -1 R * (α n ) by Φ -1 R 1 -2 + (x) (1 -α n ) -1 -2 -1
. The idea is then to estimate a quantile of level

1 -v n = 1 -2 + (x) (1 -α n ) -1 -2 -1
on the unconditional radius R, easier to deal with. By noticing that nv n ∼ (x) -1 n(1 -α n ) → +∞ as n → +∞, we introduce the following statistic order based estimator qαn (Y |X = x) for q αn (Y |X = x), inspired by Theorem 2.4.1 in de [START_REF] De Haan | Extreme value theory: an introduction[END_REF].

Definition 4.1 (Intermediate quantile estimator). We define (q αn (Y |X = x)) n∈N as :

(4.1) qαn (Y |X = x) = µ Y |X + σ Y |X W [nṽn+1] 1 ηkn ,
where ṽn = 2 + ˆ kn,hn (x)

1 1-αn -2 -1
, ηkn and ˆ kn,hn (x) are respectively given in Definitions 3.1 and 3.3, and W is the first (or indifferently any) component of the vector Λ X -1 (X -µ X ).

In order to prove the consistency of our estimator, we need a further condition (C int ) concerning the sequences α n and k n , usefull in the proof.

• (C int ) : n(1 -α n ) → +∞, ln(1 -α n ) = o( √ k n ) and √ kn ln(1-αn) = o n(1 -α n ) as n → +∞. Obviously, (C int ) contains n(1 -α n ) → +∞, as mentioned above. Furthermore, ln(1 -α n ) = o( √ k n )
ensures that the rate of convergence in Theorem 4.1 goes to infinity (see below) and the last relationship allows us to eliminate a term in the proof. In order to make this condition more meaningful, let us propose a simple example: we choose our sequences in polynomial forms k n = n b , 0 < b < 1 and

α n = 1 -n -a , a > 0. It is straightforward to see that ln(1 -α n ) = o(k n ) and ln(n(1 -α n )) = o(k n ) , ∀a > 0, 0 < b < 1. However, √ kn ln(1-αn) = o n(1 -α n ) if and only if a < 1, i.e n(1 -α n ) → +∞ as n → +∞.
In a first time, we give a result concerning the asymptotic behavior of qαn (Y |X = x) with respect to q αn↑ (Y |X = x). Then, with Equation (2.14), we easily deduce a consistency result for qαn (Y |X = x).

Theorem 4.1 (Consistency of qαn (Y |X = x)). Let us denote v n = 2 + (x) (1 -α n ) -1 -2 -1 and ṽn = 2 + ˆ kn,hn (x) (1 -α n ) -1 -2 -1
. Under Assumption 1, and conditions (C), (C int ) :

(4.2) √ k n ln (1 -α n ) qαn (Y |X = x) q αn↑ (Y |X = x) -1 → n→+∞ N 0, N 2 γ 4 (γN + 1) 4 .
And therefore :

(4.3) qαn (Y |X = x) q αn (Y |X = x) P → 1.
The same asymptotic normality with

Φ -1 R * (α n ) instead of Φ -1 R (1 -v n ) 1
η may be deduced from Proposition 4.1 under the condition

lim n→+∞ √ k n ln (1 -α n ) ln Φ -1 R (1 -v n ) 1 η Φ -1 R * (α n ) = 0.
This condition, which seems quite simple, is difficult to prove in a general context. Indeed, we need a second order expansion of Equation (2.14). But the second order properties of the unconditional quantile Φ -1 R given by Assumption 1 are not necessarily the same as those of the conditional quantile Φ -1 R * , which makes the study complicated. However, in some simple cases, we are able to solve the problem. We thus give another assumption, stronger that Assumption 1. In the following, we refer to this assumption for results of asymptotic normality.

Assumption 2. ∀d ∈ N * , there exists λ 1 , λ 2 ∈ R such that :

(4.4) c d g d (t) = λ 1 t -d+γ -1 2 1 + λ 2 t ρ 2γ + o t ρ 2γ
.

It is obvious that Assumption 2 implies Assumption 1. Indeed, according to Hua and Joe (2011), Equation (4.4) is equivalent to say that c 1 g 1 (t 2 ) is regularly varying of second order with indices -1-γ -1 , ρ/γ and an auxiliary function proportional to t ρ γ . Then, Proposition 6 in Hua and Joe (2011) entails ΦR (t) is second order regularly varying with -γ -1 , ρ/γ and the same kind of auxiliary function. Finally, this is equivalent (see de [START_REF] De Haan | Extreme value theory: an introduction[END_REF]) to Assumption 1 with indicated γ and ρ, and an auxiliary function A(t) proportional to t ρ . Furthermore, according to [START_REF] Kano | Consistency property of elliptical probability density functions[END_REF], the dependance on d in Equation (4.4) remains coherent with the assumption of consistent elliptical distributions, the latter having to have a function g d depending on d. As an example, the Student distribution fills Assumption 2. The latter allows us to provide a second order expansion for Equation (2.14). In order to prove the asymptotic normality of qαn (Y |X = x), we add a technical condition C HG int that involves tail indices γ and ρ.

•

C HG int : (C int ) holds. In addition, √ k n (1 -α n ) = o (ln(1 -α n ))
, and :

(4.5) lim n→+∞ √ k n ln (1 -α n ) (1 -α n ) min(-ρ,2γ) γN +1 = 0.
Condition C HG int means that sequence k n must not be too large. In view of Equation (4.5), it is obvious that if N or γ goes to infinity, C HG int is not filled. The tail of the underlying distribution may thus not be too heavy, and the size N of the covariate not too large. Similarly, they no longer hold if γ or ρ goes to 0, i.e. if the underlying distribution is either too lightly varying, or its c.d.f. takes too long to behave like λt -1/γ . Proposition 4.2 (Asymptotic normality of qαn (Y |X = x)). Assume that Assumption 2 and conditions (C), C HG int hold. Then :

(4.6) √ k n ln (1 -α n ) qαn (Y |X = x) q αn (Y |X = x) -1 → n→+∞ N 0, N 2 γ 4 (γN + 1) 4 .
We notice that asymptotic variance in Equation (4.2) tends to 0 as the number of covariates N goes to +∞. Indeed, we observe a fast convergence of qαn to q αn↑ when N is large. However, C HG int is not filled if N is tall. Then asymptotic normality (4.6) no longer holds. This is explained by the fact that more N is tall, more q αn (Y |X = x) /q αn↑ (Y |X = x) (see Equation (2.14)) tends to 1 slowly. 4.2. High quantiles. We now consider n(1 -α n ) → 0 as n → +∞. In the following definition, we introduce another quantile estimator qαn (Y |X = x) for q αn (Y |X = x). We first recall that the idea is to estimate an unconditional quantile of level 1

-v n = 1 -2 + (x) (1 -α n ) -1 -2 -1 . A quick calculation proves that v n is asymptotically equivalent to (x) -1 (1 -α n )
, and therefore nv n → 0 as n → +∞. The use of statistic order (at level nv n ) is then impossible in that case. According to Theorem 4.3.8 in de [START_REF] De Haan | Extreme value theory: an introduction[END_REF], a way to estimate such a quantile may be to take the statistic order at the intermediate level k n (we recall k n → +∞), and apply an extrapolation coefficient (k n /(nv n )) γ . This approach inspired the following estimator.

(4.7) qαn (Y |X = x) = µ Y |X + σ Y |X W [kn+1] k n n 2 + ˆ kn,hn (x) 1 1 -α n -2 γkn 1 ηkn .
The aim is now to study the asymptotic properties of qαn (Y |X = x). As for the intermediate quantile estimator, we propose a result of asymptotic normality, under a condition (C high ) (given below) which we then refine under Assumption 2.

• (C high ) : n(1 -α n ) → 0, ln (n(1 -α n )) = o( √ k n ) and ln(1-αn) ln( n kn (1-αn)) → θ ∈ [0, +∞[ as n → +∞.
The second statement is added in order to apply Theorem 4.3.8 in de [START_REF] De Haan | Extreme value theory: an introduction[END_REF], and the third one is a notation used in the following. Let us propose a simple example: if we choose our sequences in polynomial forms

k n = n b , 0 < b < 1 and α n = 1 -n -a , a > 0, the first condition is filled if and only if a > 1, ln(n(1 -α n )) = o( √ k n )
and the last assertion holds with a particular θ given later. The consistency result that follows immediatly is given just below.

Theorem 4.3 (Consistency of high quantile estimator). Let us denote

v n = 2 + (x) (1 -α n ) -1 -2 -1 and ṽn = 2 + ˆ kn,hn (x) (1 -α n ) -1 -2 -1
. Under Assumption 1, and conditions (C), (C high ) :

(4.8) √ k n ln kn n(1-αn) qαn (Y |X = x) q αn↑ (Y |X = x) -1 → n→+∞ N 0, γ γN + 1 -θ N γ 2 (γN + 1) 2 2 .
And therefore :

(4.9) qαn (Y |X = x) q αn (Y |X = x) P → 1 as n → +∞.
We can emphasize that condition (C high ) is filled in most of the common cases. Indeed, the simple examples to find that do not satisfy (ii) are of the form α n = 1 -n -1 ln(n) -κ , κ > 0 and k n = ln(n). But such a choice of sequences would lead to a poor estimation of γkn and ηkn , since k n → +∞ very slowly, and moreover a poor estimation of the quantile, the level α n tending to 1 slowly. These sequences are therefore not recommanded in practice. Next corollary gives the value of θ when sequences k n and α n have a polynomial form.

Corollary 4.4. Under Assumption 1, conditions (C), (C high ), and taking k n = n b , 0 < b < 1 and α n = 1 -n -a , a > 1, asymptotic relationship (4.8) holds with θ = a a+b-1 . As for the intermediate quantile estimator, asymptotic normality (4.8) may be improved under the condition

lim n→+∞ √ k n ln kn n(1-αn) ln Φ -1 R (1 -v n ) 1 η Φ -1 R * (α n ) = 0
Assumption 2 places us in a framework where it is quite simple to prove it, if we add the following condition :

• C HG high : (C high ) holds. In addition, (4.10)

lim n→+∞ √ k n ln kn n(1-αn) (1 -α n ) min(-ρ,2γ) γN +1 = 0.
As C HG int , condition C HG high means that sequence k n must be small enough. In view of Equation (4.10), we deduce that if N or γ goes to infinity, C HG high is not filled. The tail of the underlying distribution may thus not be too heavy, and the size N of the covariate not too large. Similarly, they no longer hold if γ or ρ goes to 0. By combining Assumption 2 and C HG high , the following result is obtained.

Proposition 4.5 (Asymptotic normality of high quantile estimator). Assume that Assumption 2 and conditions (C), C HG high hold. Then :

(4.11) √ k n ln kn n(1-αn) qαn (Y |X = x) q αn (Y |X = x) -1 → n→+∞ N 0, γ γN + 1 -θ N γ 2 (γN + 1) 2 2 .
We can make the same kind of remark as in the previous subsection when N is large. In the following, we give estimators for two other classes of extreme risk measures, based on the estimators given in Equations (4.1) and (4.7). The first one generalizes quantiles.

5. Some extreme risk measures estimators 5.1. L p -quantiles. Let Z be a real random variable. The L p -quantiles of Z with level α ∈]0, 1[ and p > 0, denoted q p,α (Z), is solution of the minimization problem (see [START_REF] Chen | Conditional Lp-quantiles and their application to the testing of symmetry in nonparametric regression[END_REF]) :

(5.1)

q p,α (Z) = arg min z∈R E (1 -α) (z -Z) p + + α (Z -z) p + ,
where Z + = Z1 {Z>0} . According to [START_REF] Koenker | Regression quantiles[END_REF], the case p = 1 leads to the quantile q 1,α (Z) = F -1 Z (α), where F Z is the c.d.f of Z. The case p = 2, formalized in [START_REF] Newey | Asymmetric least squares estimation and testing[END_REF], leads to more complicated calculations, and admits, with the exception of some particular cases (see, e.g., [START_REF] Koenker | When are expectiles percentiles?[END_REF]), no general formula. The general case p ≥ 1 has seen some recent advances. [START_REF] Bellini | Generalized quantiles as risk measures[END_REF] has shown that L p -quantiles get the translation equivariance and positively homogeneity properties for p > 1. More recently, the particular case of Student distributions has, for example, been explored in [START_REF] Bernardi | On the Lp-quantiles for the Student t distribution[END_REF]. However, it seems difficult to obtain a general formula. On the other hand, in the case of extreme levels α, i.e. when α tends to 1, [START_REF] Daouia | Extreme M-quantiles as risk measures: from L1 to Lp optimization[END_REF] proved that the following relationship holds, for a heavy-tailed random variable with tail index γ.

(5.2)

q p,α (Z) q α (Z) → α→1 γ B (p, γ -1 -p + 1) -γ := f L (γ, p) ,
where B(., .) is the beta function. We add that for a Pareto-type distribution with tail index γ, the L p -quantile exists if and the only if the moment of order p -1 exists, i.e. if γ < 1/p. The expectile case p = 2 leads to the result of [START_REF] Bellini | Generalized quantiles as risk measures[END_REF]. Using this result, we can estimate the conditional L p -quantiles from the quantile estimated in Section 4. For that purpose, we need to know the tail index of the conditional radius R * , given in the following lemma.

Lemma 5.1. The conditional distribution Y |X = x is attracted to a maximum domain of Pareto-type distribution with tail index (γ -1 + N ) -1 , i.e

(5.3) lim t→+∞ ΦR * (ωt) ΦR * (t) = ω -1 γ -N .
With Lemma 5.1 and Equation (5.2), we define the following estimators for the L p -quantile of Y |X = x, according to whether if n(1 -α n ) tends to 0 or +∞. Definition 5.1. Let (α n ) n∈N be a sequence such that α n → 1 as n → +∞. If either p ≤ N or γ < 1 p-N , we define:

(5.4)

     qp,αn (Y |X = x) = µ Y |X + σ Y |X W [nṽn+1] 1/η kn f L γ-1 kn + N -1 , p qp,αn (Y |X = x) = µ Y |X + σ Y |X W [kn+1] kn nṽn γkn 1/η kn f L γ-1 kn + N -1 , p .
where γkn and ṽn are respectively given in Equation (3.1) and Theorem 4.1.

We have proved the convergence in probability of qαn (Y |X = x) and qαn (Y |X = x). Furthermore, the convergence in probability of the asymptotic term, and consequently the empirical L p -quantile is not difficult to get, this is why we omit the proof. (5.5)

   qp,αn (Y |X=x) qp,α n (Y |X=x) P → 1 qp,αn (Y |X=x) qp,α n (Y |X=x) P → 1 .
Using the second order expansion of Equation (5.2) given in [START_REF] Daouia | Extreme M-quantiles as risk measures: from L1 to Lp optimization[END_REF], and doing some stronger assumptions, we can deduce the following asymptotic normality results. For that purpose, let us add two conditions.

• C Lp int : (C int ) holds. In addition,

√ k n (1 -α n ) = o (ln(1 -α n ))
, and :

(5.6) lim

n→+∞ √ k n ln (1 -α n ) (1 -α n ) min(-ρ,γ) γN +1 = 0.
• C (5.8)

     √ kn ln(1-αn) qp,αn (Y |X=x) qp,α n (Y |X=x) -1 → n→+∞ N 0, N 2 γ 4 (γN +1) 4 √ kn ln( kn n(1-αn) ) qp,αn (Y |X=x) qp,α n (Y |X=x) -1 → n→+∞ N 0, γ γN +1 -θ N γ 2 (γN +1) 2 2 .
An example of L 2 -quantile, or expectile, is provided in Section 6. The second risk measure we focus on is called Haezendonck-Goovaerts risk measure. 5.2. Haezendonck-Goovaerts risk measures. Let Z be a real random variable, and ϕ a non negative and convex function with ϕ(0) = 0, ϕ(1) = 1 and ϕ(+∞) = +∞. The Haezendonck-Goovaerts risk measure of Z with level α ∈]0, 1[ associated to ϕ, is given by the following (see [START_REF] Tang | On the Haezendonck-Goovaerts risk measure for extreme risks[END_REF]) :

(5.9)

H α (Z) = inf z∈R {z + H α (Z, z)} ,
where H α (Z, z) is the unique solution h to the equation :

(5.10)

E ϕ (Z -z) + h = 1 -α.
ϕ is called Young function. This family of risk measures has been firstly introduced as Orlicz risk measure in [START_REF] Haezendonck | A new premium calculation principle based on Orlicz norms[END_REF], then Haezendonck risk measure in [START_REF] Goovaerts | Some new classes of consistent risk measures[END_REF], and finally Haezendonck-Goovaerts risk measure in [START_REF] Tang | On the Haezendonck-Goovaerts risk measure for extreme risks[END_REF]. According to [START_REF] Bellini | On Haezendonck risk measures[END_REF], such a risk measure is coherent, and therefore translation equivariant and positively homogeneous. The particular case ϕ(t) = t leads to the Tail Value at Risk with level α TVaR α (X), introduced in [START_REF] Artzner | Coherent measures of risk[END_REF]. In the following, we denote H p,α (Z) the Haezendonck-Goovaerts risk measure of Z with a power Young function t p , p ≥ 1. In [START_REF] Tang | On the Haezendonck-Goovaerts risk measure for extreme risks[END_REF], the authors provided the following result.

Proposition 5.4 [START_REF] Tang | On the Haezendonck-Goovaerts risk measure for extreme risks[END_REF]). If Z fills Assumption 1, and taking a Young function ϕ(t) = t p , p ≥ 1, then the following relationship holds :

(5.11)

H p,α (Z) q α (Z) → α→1 γ -1 γ -1 -p pγ-1 p γ(p-1) B γ -1 -p, p γ := f H (γ, p) .
In particular, taking p = 1 leads to TVaR α (Z) ∼ (1-γ) -1 q α (Z) as α → 1. Using Lemma 5.1, extreme quantiles estimators in Definitions 4.1, 4.2 and Proposition 5.4, we can deduce estimators for extreme Haezendonck-Goovaerts risk measure

H p,α (Y |X = x) (with power Young function ϕ(t) = t p , p ≥ 1) of Y |X = x.
Definition 5.2. Let (α n ) n∈N be a sequence such that α n → 1 as n → +∞. If either p ≤ N or γ < 1 p-N , we define :

(5.12)

     Ĥp,αn (Y |X = x) = µ Y |X + σ Y |X W [nṽn+1] 1/η kn f H γ-1 kn + N -1 , p Ĥp,αn (Y |X = x) = µ Y |X + σ Y |X W [kn+1] kn nṽn γkn 1/η kn f H γ-1 kn + N -1 , p .
The condition p ≤ N or γ < 1 p-N simply ensures the existence of H p,αn (Y |X = x). Using the consistency results given in Propositions 4.1 and 4.3, the consistency of these estimators is immediate. The proof is also omitted from the appendix. (5.14)

     √ kn ln(1-αn) Ĥp,αn (Y |X=x) Hp,α n (Y |X=x) -1 → n→+∞ N 0, N 2 γ 4 (γN +1) 4 √ kn ln( kn n(1-αn ) ) Ĥp,αn (Y |X=x) Hp,α n (Y |X=x) -1 → n→+∞ N 0, γ γN +1 -θ N γ 2 (γN +1) 2 2 .
We can emphasize that conditions for asymptotic normality are less strong in the case of Haezendonck-Goovaerts risk measures. We propose some examples (with p = 1, i.e TVaR) in Sections 6 and 7.

Simulation study

In this section, we apply our estimators to 100 samples of n simulations of a Student vector Z = (X, Y ) ∈ R 4 (X ∈ R 3 and Y ∈ R) with ν = 2 degrees of freedom, and compare with theoretical results. According to de [START_REF] De Haan | Extreme value theory: an introduction[END_REF], the Student distribution with ν degrees of freedom fills Assumption 1 with indices γ = 1/ν, ρ = -2/ν, and an auxiliary function A(t) proportional to t -2/ν . The latter even fills Assumption 2, and is the only heavy-tailed elliptical distribution (to our knowledge) where we can obtain closed formula for conditional quantiles. In addition, such a degree of freedom makes the tail of the distribution sufficiently heavy to easily observe the asymptotic results. We can notice that the unconditional distribution Y has tail index 1/2, then, using Lemma 5.1, the conditional distribution Y |X = x has tail index 2/7 < 1/2, and admits quantile, expectile (L 2 -quantile) and TVaR. This section beeing uniquely devoted to the performance of our estimators, we take for conveniance µ = 0 R 4 and Σ = I 4 . Let us now estimate the extreme quantiles of Y |X = x. For that purpose, we have to chose an arbitrary value of x. We thus suppose for example that the observed covariates x satisfy M (x) = 1. 6.1. Choice of parameters. As mentioned in Sections 3 and 4, the asymptotic results obtained are sensitive to the choice of sequences k n , h n , α n , and to a lesser extent to the kernel K. The latter will be the gaussian p.d.f in the following. Concerning the sequences, we propose in this section to consider the polynomial forms α n = 1 -n -a , a > 0, k n = n b , b > 0 and h n = n -c , c > 0. In order to deal with high quantiles, we fix in a first time a = 1. For that purpose, we use our estimators ηkn and ˆ kn,hn (x) respectively introduced in Equations ( 3.3) and (3.10). These two estimators are related to the Hill estimator γkn , and asymptotic results of Section 3 hold only if the data is independent. This is why we do the estimation of γ only with the n realizations of the first component from the vector Z. Figure 1 shows the boxplots of our estimators ηkn and ˆ kn,hn (x). In this example, the theoretical value of η is 3/2 + 1 = 2.5, and (x) is equal to 5.292757 (cf. Table 1). 6.3. Extreme risk measures estimation. It remains to estimate the conditional quantiles, expectiles and TVaRs of Y |X = x. Theoretical formulas (or algorithm) for conditional quantiles and expectiles may be found in [START_REF] Maume-Deschamps | Quantile predictions for elliptical random fields[END_REF] and [START_REF] Maume-Deschamps | Spatial expectile predictions for elliptical random fields[END_REF]. Furthermore, using straightforward calculations, formulas for Tail-Value-at-Risk may be obtained. (6.1)

     q α (Y |X = x) = ν+M (x) ν+N Φ -1 ν+N (α) TVaR α (Y |X = x) = 1 1-α Γ( N +1+ν 2 ) Γ( N +ν 2 ) √ ν+M (x) √ π(ν+N -1) 1 + Φ -1 ν+N (α) 2 ν+N 1-N -ν 2
, where Φ ν is the c.d.f of a Student distribution with ν degrees of freedom. In order to give an idea of the performance of our estimator, we propose in Figure 2 some box plots representing 100 relative errors (based on sample sizes n from 1 000 to 10 000 000) of our quantile estimator (4.7) with α n = 1 -n -1.25 . Finally, we would like to compare these results with other estimators already used. The most common and widespread methods for estimating conditional quantiles and expectiles are respectively quantile and expectile regression, introduced in [START_REF] Koenker | Regression quantiles[END_REF] and [START_REF] Newey | Asymmetric least squares estimation and testing[END_REF]. In Maume-Deschamps et al. ( 2017) and Maume-Deschamps et al. ( 2018), we have shown that such approach leads to a poor estimation in case of extreme levels. Indeed, in this example, a quantile regression estimator will converge to Φ -1 ν (α n ) = 1530.15, very far from 7.31, the theoretical result. Obviously, since the quantile regression estimator does not assume any structure on the underlying distribution, the latter is clearly less efficient than the tailored extreme quantile estimators introduced in this paper. qα n (Y |X=x) -1 (based on sample sizes n from 1 000 to 10 000 000) with α n = 1 -n -1.25 , k n = n 0.6 and h n = n -0.2 . It may also be interesting to compare the empirical variance of our estimator with our asymptotic result given in Proposition 4.5. Furthermore, the latter allow us to provide confidence intervals for q αn (Y |X = x). We thus introduce the notation ζn the empirical variance of

√ kn ln( kn n(1-αn ) ) qαn (Y |X=x) qα n (Y |X=x) -1 , while ζ = γ γN +1 -θ N γ 2 (γN +1) 2 2
= 0.0005536332 in this section. In addition, we denote m n the number of times the theoretical value q αn (Y |X = x) is in the 95% confidence interval. Table 2 gives an overview of the behavior of these quantities according to n. n α n ζn m n 1 000 0.9998222 0.001839654 44 10 000 0.99999 0.001110249 76 100 000 0.9999994 0.0006940025 90 1 000 000 0.99999996837 0.0005534661 94 10 000 000 0.99999999822 0.0005695426 91 +∞ 1 0.0005536332 95

Table 2. Empirical variance ζn , number of confidence intervals containing the theoretical value m n for 100 estimations qαn (Y |X = x) of q αn (Y |X = x), with n ranging from 1 000 to 10 000 000. Chosen sequences are α n = 1 -n -1.25 , k n = n 0.6 and h n = n -0.2 .

Finally, based on these quantile estimates, we deduce, using Definitions 5.1 and 5.2, L 2 -quantile (or expectile) and Tail-Value-at-Risk estimates. Figure 3 provides relative errors for estimators q2,αn and Ĥ1,αn . H 1,αn (Y |X=x) -1 (based on sample sizes n from 1 000 to 10 000 000) with α n = 1 -n -1.25 , k n = n 0.6 and h n = n -0.2 .

In the previous figures, only the first component of the vector is used to estimate the tail index. There is therefore some loss of information. We have suggested in Section 3 another approach. Furthermore, [START_REF] Resnick | Consistency of Hill's Estimator for Dependent Data[END_REF] or [START_REF] Hsing | On Tail Index Estimation Using Dependant Data[END_REF] proved that the Hill estimator may also work with dependent data. Thus it would be possible to improve the estimation of γkn by adding the other components of the vector in Equation (3.1), but in that case the asymptotic results of Propositions 3.1 or 3.3 would not hold anymore.

Real data example

As an application, we use the daily market returns (computed from the closing prices) of financial assets from 2006 to 2016, available at http://stanford.edu/class/ee103/portfolio.html. We focus on the first four assets, i.e iShares Core U.S. Aggregate Bond ETF, PowerShares DB Commodity Index Tracking Fund, WisdomTree Europe SmallCap Dividend Fund and SPDR Dow Jones Industrial Average ETF which will be our covariate X. The reason for focusing solely on the value of these assets could be, for example, that they are the first available every day. The aim would be to anticipate the behavior of another asset on another market. We thus consider the return of WisdomTree Japan Hedged Equity Fund as random variable Y . The size of the sample is 2520. The first 2519 days (from January 3, 2007 to December 5, 2016) will be our learning sample, and we focus on the 2520th day, when the covariate X is x = (-0.0185%, -0.4464%, 0.9614%, 0.1405%). Pending the opening of the second market, let us estimate the quantile of the return Y given X = x. After a brief study of the autocorrelation functions, we consider that the daily returns can be considered as independent. Concerning the shape of the data, histograms of the marginals seem symmetrical.

Furthermore, the measured tail index is approximately the same for the 4 marginals. This is why suppose that the data is elliptical. After having estimated µ and Σ by the method of moments, we get M (x) = 1.072952. We apply our estimators ηkn and ˆ kn,hn (x) given in Equations (3.3) and (3.10). We take as sequences k n = n 0.6 (b = 0.6) and h n = n -0.2 (c = 0.2), and as kernel K the gaussian p.d.f, hence we deduce the asymptotic confidence bounds from Equation (3.14). We then obtain ηkn = 2.617846 and ˆ kn,hn (x) = 6.44334. Let us now estimate the high quantile q αn (Y |X = x) with level α n = 1-n -a , a > 1. In order to minimize the asymptotic variance of Equation (4.11), we chose a = (1-b) (γ kn + 1) = 1.047146. By applying estimator (4.7), we get a quantile of level 0.9997256 close to 3.744985% for Y |X = x. In other words, before the opening of the second market, we consider that given the returns of our first four assets, that of WisdomTree Japan Hedged Equity Fund has a probability 0.9997256 of beeing less than 3.744985%. For information, the true return that day was 0.7141%.

Conclusion

In this paper, we propose two estimators ˆ kn,hn (x) and ηkn respectively for extremal parameters (x) and η introduced in Equation (2.13). We have proved their consistency and asymptotic normality according to the asymptotic relationships between the sequences k n and h n . Using these estimators, we have defined estimators for intermediate and high quantiles, proved their consistency, given their asymptotic normality under stronger conditions, and deduced estimators for extreme L p -quantiles and Haezendonck-Goovaerts risk measures. Consistency and asymptotic normality are also provided for these estimators, under conditions. We have also illustrated with a numerical example the performance of our estimators, and applied them to real data set. As working perspectives, we intend to propose a method of optimal choice of the sequences k n and h n , which is not totally discussed in this paper. Furthermore, the shape of (x) and η leaving Assumption 1 is a current research topic. More generally, the asymptotic relationships between conditional and unconditional quantile in other maximum domains of attraction, using for example the results of Hashorva (2007a), may be developed. However, we need a second-order refinement, as we need a second-order refinement of Equation (2.14) to propose asymptotic normalities 4.2 and 4.5 under weaker assumptions than Assumption 2. Finally, it seems that the ratio of the two terms in Equation (2.14) tends to 1 more and more slowly when the covariate vector size N becomes large. Then, our estimation approach may perform poorly if N is tall. This is why it might be wise to propose another method when the covariate vector size N is large.

Appendix

Proof of Lemma 2.1. [START_REF] Jessen | Regularly varying functions[END_REF], ξ satisfies Fξ (tω)/ Fξ (t) → ω -1 γ as t → +∞. Furthermore, Lemma 4.2 in [START_REF] Jessen | Regularly varying functions[END_REF] entails

(i) Since R 1 d = χ 1 ξ,
P (ξ > t) ∼ t→+∞ E χ 1 γ 1 -1 P (R 1 > t) .
Assumption 1 provides P(R 1 > t) ∼ λt -1 γ , hence the result. (ii) Using again Lemma 4.2 in [START_REF] Jessen | Regularly varying functions[END_REF]

for R d d = χ d ξ, it comes immediatly P (R d > t) ∼ t→+∞ E χ 1 γ d P (ξ > t) . Some straightforward calculations provide E χ 1 γ d = 2 1 γ Γ d+γ -1 2 Γ( d 2 ) . (iii) From (ii), we have, for all d ∈ N, f R d (t) ∼ t→+∞ 2 1 γ Γ d+γ -1 2 Γ( d 2 ) λ t -1 γ -1 ,
lim t→∞ ΦR * (t) ΦR (t η ) = lim t→∞ c N +1 g N +1 (M (x) + t 2 ) c N g N (M (x)) ηt η-1 c 1 g 1 (t 2η ) = lim t→∞ Γ N +1 2 (M (x) + t 2 ) -N 2 π N +1 2 c N g N (M (x)) ηt η-1 f R N +1 M (x) + t 2 f R1 (t η ) .
Using Equation (2.11) of Lemma 2.1, it comes

ΦR * (t) ΦR (t η ) ∼ t→+∞ 1 π N 2 c N g N (M (x)) η Γ N +1+γ -1 2 Γ 1+γ -1 2 t (η-1)(γ -1 +1)+1-η-N .
Obviously, we impose 0 < (x) < +∞, then 1 -η -N + (η -1)(γ -1 + 1) = 0, hence η = N γ + 1. Replacing η in the previous equation, (x) is easily deduced :

(x) = 1 π N 2 c N g N (M (x)) η Γ N +1+γ -1 2 Γ 1+γ -1 2 . 2 Proof of Proposition 3.4. It is obvious that under conditions (K1)-(K2), √ k n (ĝ hn -c N g N (M (x))) P → 0 as n → +∞ if k n = o(nh n ) and √ k n h 2 n → 0.
Then we get the following asymptotic normality :

k n γkn -γ ĝhn -c N g N (M (x)) → n→+∞ N 0 0 , γ 2 0 0 0 .
Since (x) = u(γ), the delta method entails

k n ˆ kn,hn (x) -(x) → n→+∞ N 0, u (γ) 2 γ 2 .
A quick calculation of u , using Equation (2.12), gives the first result. The second part of the proof is similar. Indeed, if nh n = o(k n ) and nh 5 n → 0 as n → +∞, then

nh n γkn -γ ĝhn -c N g N (M (x)) → n→+∞ N   0 0 ,   0 0 0 M (x) 1-N 2 Γ( N 2 ) π N 2 c N g N (M (x)) K(u) 2 du     .
The delta method completes the proof. 2

In order to make the proof of Theorem 4.1 easier to read, we give the following lemma, which provides the asymptotic behavior of a statistic order under Assumption 1.

Lemma 8.1. Under Assumption 1 and condition (C), (8.1)

√ nv n W [nvn+1] Φ -1 R (1 -v n ) -1 → n→+∞ N 0, γ 2 .
Proof of Lemma 8.1. The proof is inspired by Theorem 2.4.1 in de [START_REF] De Haan | Extreme value theory: an introduction[END_REF]. Let Y 1 , Y 2 , . . . be independant and identically distributed random variables with c.d.f. 1 -y -1 , y > 1. We denote in addition

Y [n] ≤ . . . ≤ Y [1] . We thus have √ nv n v n Y [nvn+1] -1 → n→+∞ N (0, 1). By noticing that W [nvn+1] d = Φ -1 R 1 -1/Y [nvn+1] , it comes √ nv n W [nvn+1] Φ -1 R (1 -v n ) -1 d = √ nv n Φ -1 R 1 -1/Y [nvn+1] Φ -1 R (1 -v n ) -v n Y [nvn+1] γ + √ nv n v n Y [nvn+1] γ -1 .
The delta method entails that the second term tends to N (0, γ 2 ). Moreover, Assumption 1 and √ k n A n kn → 0 as n → +∞ ensure the asymptotic nullity of the first term.

Proof of Theorem 4.1. In a first time, we can notice ṽn is related to ˆ kn,hn (x). Then, according to Proposition 3.4, (i) entails that we can deal with v n instead of ṽn in Equation (4.2). Furthermore, we give the decomposition :

√ k n ln (1 -α n ) qαn (Y |X = x) q αn↑ (Y |X = x) -1 ∼ n→+∞ √ k n ln (1 -α n ) W [nvn+1] 1/η kn Φ -1 R (1 -v n ) 1/η -1 = √ k n ln (1 -α n ) W [nvn+1] 1/η kn Φ -1 R (1 -v n ) 1/η kn -1 Φ -1 R (1 -v n ) 1/η kn -1/η + √ k n ln (1 -α n ) Φ -1 R (1 -v n ) 1/η kn -1/η -1 .
Under Assumption 1, and according to Proposition 3.1 and Theorem 2.4.1 in de [START_REF] De Haan | Extreme value theory: an introduction[END_REF] (with (C)), we have :

     √ k n 1 ηkn -1 η → n→+∞ N 0, N 2 γ 2 (γN +1) 4 √ nv n (W[nv n +1]) Φ -1 R (1-vn) -1 → n→+∞ N 0, γ 2
, By noticing that v n is equivalent to (x) -1 (1-α n ) as n → +∞, and using condition

√ k n = o ln(1 -α n ) n(1 -α n ) in (C int ), it comes √ k n ln (1 -α n ) W [nvn+1] 1/η kn Φ -1 R (1 -v n ) 1/η kn -1 → 0 as n → +∞. Furthermore, under Assumption 1, ln Φ -1 R (1 -v n ) is cleary equivalent to -γ ln(v n ), or -γ ln(1 -α n ). Then (C int ) ensures Φ -1 R (1 -v n )
1/η kn -1/η → 1 as n → +∞, and therefore the first term of the decomposition tends to 0. It thus remains to calculate the limit of the second term. It is not complicated to notice that

√ k n ln Φ -1 R (1 -v n ) Φ -1 R (1 -v n ) 1/η kn -1/η -1 → n→+∞ N 0, N 2 γ 2 (γN + 1) 4 .
Using the equivalence ln Φ

-1 R (1 -v n ) ∼ -γ ln(v n ) ∼ -γ ln(1 -α n )
, we get the result (4.2). Using asymptotic relationship (2.14), the consistency 4.3 is obvious. 2 Proof of Proposition 4.2. We recall that density of Φ R * is proportional to c N +1 g N +1 M (x) + t 2 , and, from Assumption 2, there exist λ 1 , λ 2 ∈ R such that :

c N +1 g N +1 M (x) + t 2 = λ 1 M (x) + t 2 -N +1+γ -1 2 1 + λ 2 M (x) + t 2 ρ 2γ + o t ρ γ
The previous expression may be rewritten as follows, where λ 1 , λ 2 , λ 3 ∈ R :

c N +1 g N +1 M (x) + t 2 = λ 1 t -(N +1+γ -1 ) 1 + λ 2 M (x) + t 2 ρ 2γ + λ 3 t -2 + o t ρ γ
In order to make the proof more readable, we do not specify the values of constants λ i , because they are not essential. Then, in the case, ρ/γ ≤ -2, we get

c N +1 g N +1 M (x) + t 2 = λ 1 t -(N +1+γ -1 ) 1 + λ 2 t -2 + o t -2 , λ 1 , λ 2 ∈ R
In other terms, c N +1 g N +1 M (x) + t 2 is regularly varying of second order with indices -N -1 -γ -1 , -2, and an auxiliary function propotional to t -2 . According to Proposition 6 of [START_REF] Hua | Second order regular variation and conditional tail expectation of multiple risks[END_REF]

, ΦR * (t) = +∞ t c N +1 g N +1 M (x) + u 2 du ∈ 2RV -N -γ -1 ,-2 with an auxiliary function proportional to t -2 . Equivalently, there exists λ 1 , λ 2 ∈ R such that Φ -1 R * 1 - 1 t = λ 1 t γ γN +1 1 + λ 2 t -2γ γN +1 + o t -2γ γN +1
.

Since Assumption 1 and Assumption 2 provide Φ

-1 R (1 -1/t) = λ 3 t γ [1 + λ 4 t ρ + o (t ρ )], it comes Φ -1 R (1 -v n ) 1 η Φ -1 R * (α n ) = (x) -γ γN +1 1 -α n v n γ γN +1 1 + λ 1 v -ρ n + o (v -ρ n ) 1 + λ 2 (1 -α n ) 2γ γN +1 + o (1 -α n ) 2γ γN +1
, for some constants λ 1 , λ 2 ∈ R. In that case, we considered ρ ≤ -2γ, hence -ρ > 2γ/(γN + 1). We then deduce the following expansion :

Φ -1 R (1 -v n ) 1 η Φ -1 R * (α n ) = (x) -γ γN +1 1 -α n v n γ γN +1 1 + λ(1 -α n ) 2γ γN +1 + o (1 -α n ) 2γ γN +1 , for a certain constant λ ∈ R. We can notice that (1 -α n )/v n = 2(1 -(x))(1 -α n ) + (x)
, and let us now focus on the limit :

lim n→+∞ √ k n ln (1 -α n ) ln Φ -1 R (1 -v n ) 1 η Φ -1 R * (α n ) = γ γN + 1 lim n→+∞ √ k n ln (1 -α n ) ln 2 1 -(x) (x) (1 -α n ) + 1 + lim n→+∞ √ k n ln (1 -α n ) ln 1 + λ(1 -α n ) 2γ γN +1 + o (1 -α n ) 2γ γN +1 .
The first term gives is easy to calculate. Indeed, since

√ k n (1 -α n )/ ln (1 -α n ) → 0 as n → +∞, we deduce lim n→+∞ √ k n ln (1 -α n ) ln 2 1 -(x) (x) (1 -α n ) + 1 = 2 1 -(x) (x) lim n→+∞ √ k n ln (1 -α n ) (1 -α n ) = 0.
By a similar calculation, the second term also tends to 0, supposing

√ kn ln(1-αn) (1 -α n ) 2γ
γN +1 → 0 as n → +∞. Then, we deduce, using Proposition 4.1 :

√ k n ln (1 -α n ) qαn (Y |X = x) q αn (Y |X = x) -1 ∼ √ k n ln (1 -α n ) W [nvn+1] 1/η kn Φ -1 R * (α n ) -1 = √ k n ln (1 -α n ) qαn R * U (1) Φ -1 R (1 -v n ) 1 η -1 Φ -1 R (1 -v n ) 1 η Φ -1 R * (α n ) + √ k n ln (1 -α n ) Φ -1 R (1 -v n ) 1 η Φ -1 R * (α n ) -1 → n→+∞ N 0, N 2 γ 4 (γN + 1) 4 .
Now, let us focus on the case ρ/γ > -2. The proof is exactly the same, with

c N +1 g N +1 M (x) + t 2 = λ 1 t -(N +1+γ -1 ) 1 + λ 2 t ρ γ + o t ρ γ , λ 1 , λ 2 ∈ R.
Using the same calculations and doing the further assumption lim

q αn↑ (Y |X = x) -1 ∼ n→+∞ W [kn+1] kn nṽn γkn 1 ηkn Φ -1 R (1 -v n ) 1 η -1 =      W [kn+1] kn nvn γkn 1 ηkn Φ -1 R (1 -v n ) 1 η -1      v n ṽn γkn ηkn + v n ṽn γkn ηkn -1. Since k n = o(nh n ), we deduce √ k n ln kn nvn      W [kn+1] kn nṽn γkn 1 ηkn Φ -1 R (1 -v n ) 1 η -1      ∼ n→+∞ √ k n ln kn nvn      W [kn+1] kn nvn γkn 1 ηkn Φ -1 R (1 -v n ) 1 η -1      .
Furthermore, according to Theorem 4.3.8 in de Haan and Ferreira ( 2006), (C) and (C high ) lead to

√ k n ln kn nvn    W [kn+1] kn nvn γkn Φ -1 R (1 -v n ) -1    ∼ n→+∞ √ k n ln kn nvn k n nv n γkn -γ -1 .
From Assumption 1, it is not difficult to prove that ln Φ

-1 R (1 -v n ) / ln (k n /(nv n )) is asymptotically equivalent to γ ln(1 -α n )/ ln (n(1 -α n )/k n ).
Then, if we focus on the second term, it comes, using the limit given in (C high ) :

√ k n ln kn nvn   kn nvn γkn -γ -1 Φ R (1 -v n ) 1 ηkn -1 η -1   → n→+∞ N 0 0 , γ 2 -θ N γ 3 (γN +1) 2 -θ N γ 3 (γN +1) 2 θ 2 N 2 γ 4 (γN +1) 4 . Finally, (8.3) √ k n ln kn nvn      W [kn+1] kn nṽn γkn 1 ηkn Φ -1 R (1 -v n ) 1 η -1      = √ k n ln kn nvn Φ -1 R (1 -v n ) 1 ηkn -1 η -1 + √ k n ln kn nvn      W [kn+1] kn nṽn γkn 1 ηkn Φ -1 R (1 -v n ) 1 ηkn -1      Φ -1 R (1 -v n ) 1 ηkn -1 η .
When n → ∞, this expression is the sum of the following bivariate normal distribution :

√ k n ln kn nvn    W [kn +1]( kn nṽn ) γkn 1 ηkn Φ -1 R (1-vn) 1 ηkn -1 Φ R (1 -v n ) 1 ηkn -1 η -1    → n→+∞ N 0 0 , γ 2 (γN +1) 2 -θ N γ 3 (γN +1) 3 -θ N γ 3 (γN +1) 3 θ 2 N 2 γ 4 (γN +1) 4 , To conclude, ln kn nvn ∼ ln kn n(1-αn)
as n → +∞, hence the result. The consistency is immediate. 2 Proof of Proposition 4.5. The proof is similar to that of Proposition 4.2. Indeed, we have given, in the case ρ/γ ≤ -2 :

Φ -1 R (1 -v n ) 1 η Φ -1 R * (α n ) = (x) -γ γN +1 (2(1 -(x))(1 -α n ) + (x)) γ γN +1 1 + λ(1 -α n ) 2γ γN +1 + o (1 -α n ) 2γ γN +1 , for a certain constant λ ∈ R. It thus remains to calculate lim n→+∞ √ k n ln kn n(1-αn) ln Φ -1 R (1 -v n ) 1 η Φ -1 R * (α n ) = γ γN + 1 lim n→+∞ √ k n ln kn n(1-αn) ln 2 1 -(x) (x) (1 -α n ) + 1 + lim n→+∞ √ k n ln kn n(1-αn) ln 1 + λ(1 -α n ) 2γ γN +1 + o (1 -α n ) 2γ γN +1
.

The first term is easy to calculate. Indeed, since n(1 -α n ) → 0 and k n = o(n) as n → +∞, we deduce

lim n→+∞ √ k n ln kn n(1-αn) ln 2 1 -(x) (x) (1 -α n ) + 1 = 2 1 -(x) (x) lim n→+∞ √ k n ln kn n(1-αn) (1 -α n ) = 0.
By a similar calculation, the second term also tends to 0, supposing

√ kn ln( kn n(1-αn ) ) (1 -α n ) 2γ γN +1 → 0 as n → +∞.
Then, we deduce, using Proposition 4.3 :

√ k n ln kn n(1-αn)      W [kn+1] kn nṽn γkn 1 ηkn Φ -1 R * (α n ) -1      = √ k n ln kn n(1-αn)      W [kn+1] kn nṽn γkn 1 ηkn Φ -1 R (1 -v n ) 1 η -1      Φ -1 R (1 -v n ) 1 η Φ -1 R * (α n ) + √ k n ln kn n(1-αn) Φ -1 R (1 -v n ) 1 η Φ -1 R * (α n ) -1 → n→+∞ N 0, γ 2 (γN + 1) 2 -2θ N γ 3 (γN + 1) 3 + θ 2 N 2 γ 4
(γN + 1) 4 .

Now, let us focus on the case ρ/γ > -2. The proof is exactly the same, with

Φ -1 R (1 -v n ) 1 η Φ -1 R * (α n ) = (x) -γ γN +1 (2(1 -(x))(1 -α n ) + (x)) γ γN +1 1 + λ(1 -α n ) -ρ γN +1 + o (1 -α n ) -ρ γN +1 , λ ∈ R.
Using the same calculations and doing the further assumption lim

√ k n ln (1 -α n ) qp,αn (Y |X = x) q p,αn (Y |X = x) -1 ∼ n→+∞ √ k n ln (1 -α n )   W [nṽn+1] 1/η kn f L γ-1 kn + N -1 , p q p,αn R * U (1) -1   = √ k n ln (1 -α n )   f L γ-1 kn + N -1 , p f L (γ -1 + N ) -1 , p -1   W [nṽn+1] 1/η kn Φ -1 R * (α n ) f L γ -1 + N -1 , p Φ -1 R * (α n ) q p,αn R * U (1) + √ k n ln (1 -α n ) W [nṽn+1] 1/η kn Φ -1 R * (α n ) -1 f L γ -1 + N -1 , p Φ -1 R * (α n ) q p,αn R * U (1) + √ k n ln (1 -α n )   f L γ -1 + N -1 , p Φ -1 R * (α n ) q p,αn R * U (1)
-1   .

We know that f L γ-1 kn + N -1 , p , as a function of γkn , is asymptotically normal with rate √ k n (see Equation (3.2)). Then, the first term in the sum clearly tends to 0 as n → +∞. Using Proposition 4.2, the second term tends to the normal distribution given in (4.6). Finally, we have to check that the third term tends to 0. For that purpose, we use the second order expansion given in [START_REF] Daouia | Extreme M-quantiles as risk measures: from L1 to Lp optimization[END_REF]:

q p,αn R * U (1)
f L (γ -1 + N ) -1 , p q αn R * U (1) = 1 -(γ -1 + N ) -1 r(α n , p) + (λ + o(1)) A * 1 1 -α n ,

where r(α n , p) = λ 1 1 qα n (R * U (1) ) E R * U (1) + o(1) + λ 2 A * 1 1-αn (1 + o(1)), λ, λ 1 , λ 2 ∈ R are not related to n and A * (t) is the auxiliary function of Φ R * 1 -1 t . It seems important to precise that the conditional distribution R * U (1) is regularly varying with tail index γ -1 + N > 1, then E R * U (1) exists and, R * U (1) being symmetric, equals 0. Then, a sufficient condition for asymptotic normality may be We know, using Assumption 2 and the proof of Proposition 4.5, that q αn R * U (1) = Φ -1 R * (α n ) is asymptotically proportional to (1-α n ) -γ γN +1 , while A * and using Proposition 4.5 instead of 4.2. 2 Proof of Proposition 5.6. We have the following decomposition :

√ k n ln (1 -α n ) Ĥp,αn (Y |X = x) H p,αn (Y |X = x) -1 ∼ n→+∞ √ k n ln (1 -α n )   W [nṽn+1] 1/η kn f H γ-1 kn + N -1 , p H p,αn R * U (1) -1   = √ k n ln (1 -α n )   f H γ-1 kn + N -1 , p f H (γ -1 + N ) -1 , p -1   W [nṽn+1] 1/η kn Φ -1 R * (α n ) f H γ -1 + N -1 , p Φ -1 R * (α n ) H p,αn R * U (1) + √ k n ln (1 -α n ) W [nṽn+1] 1/η kn Φ -1 R * (α n ) -1 f H γ -1 + N -1 , p Φ -1 R * (α n ) H αn R * U (1) + √ k n ln (1 -α n )   f H γ -1 + N -1 , p Φ -1 R * (α n ) H p,αn R * U (1) -1   .
We know that f H γ-1 kn + N -1 , p , as a function of γkn , is asymptotically normal with rate √ k n (see Equation (3.2)). Then, the first term in the sum clearly tends to 0 as n → +∞. Using Proposition 4.2, the second term tends to the normal distribution given in (4.6). Finally, we have to check that the third term tends to 0. For that purpose, we use the result of Theorem 4.5 in [START_REF] Mao | Second-order properties of the haezendonck-goovaerts risk measure for extreme risks[END_REF], which ensures that there exists λ ∈ R such that :

H p,αn R * U (1) f H (γ -1 + N ) -1 , p Φ -1 R * (α n ) = 1 + λA * 1 1 -α n (1 + o(1)),
where A * is the auxiliary function of Φ -1 R * 1 -1 t . In the proof of Proposition 4.2, we have seen that A * (t) was proportional either to t -2γ γN +1 if ρ ≤ -2γ or t

f H (γ -1 + N ) -1 , p Φ -1 R * (α n )   = 0.
Hence the third term in the sum tends to 0, and the first result of (5.14) is proved. The proof is exactly the same for the second one, with rate

  Definition 4.2 (High quantile estimator). We define qαn (Y |X = x) n∈N as :

  Proposition 5.2 (Consistency of L p -quantile estimators). Assume that Assumption 1 and condition (C) hold. Under conditions (C int ) and (C high ) respectively, qp,αn (Y |X = x) and qp,αn (Y |X = x) are consistent, i.e. :

  k n must be chosen smaller. Finally, we can draw the same conclusions than above, i.e. these conditions are applicable for regularly varying distributions with an intermediate level γ, and a small number of covariates N . To sum up, among all these conditions, we can deduce the following ordering : HG high ⇒ (C high ). Proposition 5.3 (Asymptotic normality of L p -quantile estimators). Assume that Assumption 2 and condition (C) hold. Under conditions C Lp int and C Lp high respectively, and if p > 1, then :

.

  Proposition 5.5 (Consistency of H-G estimators). Assume that Assumption 1 and condition (C) hold. Under conditions (C int ) and (C high ) respectively, Ĥp,αn (Y |X = x) and Ĥp,αn (Y |X = x) are consistent, iProposition 5.6 (Asymptotic normality of H-G estimators). Assume that Assumption 2 and condition (C) hold. Under conditions C HG int and C HG high respectively, we have :

  25. We now have to chose carefully the parameters b and c, fulfilling the conditions (C), (C high ) and C HG high . (C) imposes b < 1-c, b < 4c and b < 4/(ν +4) = 2/3, (C high ) is satisfied with θ = a/(a + b -1) (see Corollary 4.4), C HG high entails b ≤ 2a = 2.5 and b ≤ 4a/(N + ν) = 1. Finally, it seems reasonable to chose b (respectively c) as tall (respectively small) as possible. The choices b = 0.6 and c = 0.2 seem to be a good compromise. 6.2. Extremal parameters estimation. The next step is to estimate the quantities η and (x).

Figure 1 .

 1 Figure 1. From left to right: boxplots of 100 estimators ηkn and ˆ kn,hn (x), for different sample sizes n. Theoretical values are in red. The chosen sequences are k n = n 0.6 , h n = n -0.2 and α n = 1 -n -1.25 .

Figure 2 .

 2 Figure 2. Box plots representing 100 relative errors qαn (Y |X=x)qα n (Y |X=x) -1 (based on sample sizes n from 1 000 to 10 000 000) with α n = 1 -n -1.25 , k n = n 0.6 and h n = n -0.2 .

Figure 3 .

 3 Figure 3. From left to right : Box plots representing 100 relative errors q2,αn (Y |X=x) q 2,αn (Y |X=x) -1 and Ĥ1,αn (Y |X=x)H 1,αn (Y |X=x) -1 (based on sample sizes n from 1 000 to 10 000 000) with α n = 1 -n -1.25 , k n = n 0.6 and h n = n -0.2 .

  Figure 4 represents the daily return for each day.

Figure 4 .

 4 Figure 4. Daily market returns of 4 different assets.

  αn) (1 -α n ) -ρ γN +1 = 0 leads to the result. 2 Proof of Theorem 4.3. Firstly, we can notice that (8.2) qαn (Y |X = x)

  αn ) ) (1 -α n ) -ρ γN +1 = 0 leads to the result. 2 Proof of Lemma 5.1. The density of Y |X = x is given byc N +1 g N +1 M (x) + (t -µ Y |X ) 2 σ -2 Y |X (c N g N (M (x))) -1 , where M (x) = (x -µ X ) Σ -1 X (x -µ X ).In order to simplify, we consider the case reduced and centered, i.e µ Y |X = 0 and σ Y |X = 1. A quick calculation giveslim t→+∞ ΦR * (ωt) ΦR * (t) = ω lim t→+∞ g N +1 (M (x) + ω 2 t 2 ) g N +1 (M (x) + t 2 ) = ω lim t→+∞ (M (x) + ω 2 t 2 ) -ωω -N ω -1 γ -1 = ω -1 γ -N . 2Proof of Proposition 5.3. We recall in a first time that condition C Lp int entails C HG int . We have the following decomposition :

1 1 -

 1 αn is asymptotically proportional to (1 -α n ) -ρ γN +1 if ρ > -2γ and (1 -α n ) 2γ γN +1 otherwise.Finally, it is not difficult to check that C Lp int leads to the nullity of the two limits, and therefore to the third term of the decomposition, hence the result. The proof is exactly the same for the second normality, replacing qp,αn (Y |X = x) by qp,αn (Y |X = x), ln (1 -α n ) by ln kn n(1-αn)

  αn) . Then condition C HG high gives the expected result. 2

  where χ 1 has a Lebesgue density

	2 π e -x 2 2 . According to Lemma 4.3 in Jessen

  where λ ∈ R is not related to d. The result is immediate with this expression. 2 Proof of Proposition 2.2. The conditional density (Proposition 3 in Maume-Deschamps et al. (2017)) leads to :

Université de Lyon, Université Lyon 1, Institut Camille Jordan ICJ UMR 5208 CNRS E-mail address: usseglio@math.univ-lyon1.fr

Acknowledgements

The author would like to thank the Editor-in-Chief, the Associate Editor and the referees, who did an extremely detailed and relevant report that widely helped to improve the paper. This work was partially supported by the MultiRisk LEFE-INSU Program, and by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).