domain: math stat

text:
Laurent Gardes
email: gardes@unistra.fr

Samuel Maistre

Alex Podgorny

Asymptotic confidence intervals for extreme quantiles in a maximum domain of attraction

Keywords: Extreme quantile, confidence interval, maximum domain of attraction

Extreme quantiles are commonly used to assess the risk of extreme events such as large floods, extreme temperature, financial crisis among many others. When the underlying distribution belongs to a maximum domain of attraction, extreme quantile estimation has been widely studied in the literature but, surprisingly, little work has been dedicated to the construction of an asymptotic confidence interval. This is precisely the question addressed in this paper where a confidence interval for extreme quantile is proposed. This confidence interval can be used whatever the maximum domain of attraction of the distribution. The convergence of its coverage probability to the nominal one is established and its finite sample performance is investigated through a simulation study. An application on lifespan of French supercentenarians is done.

Introduction

Let X be a random variable defined on an atomless probability space (Ω, A, P). We denote by F X (•) := P(X ≤ •) its cumulative distribution function and by F X = 1 -F X its survival function. Throughout this paper, we assume that F X is a continuous and strictly increasing function and that the support S of the distribution is included in [0, ∞). The associated quantile of level β ∈ (0, 1) is given by

Q X (β) = F ← X (β) = inf{x ∈ S|F X (x) ≤ β}.
In various fields of application, large quantiles are commonly used to assess the risk of extreme events such as large floods, extreme temperature, financial crisis among many others. More details can be found in the monographs of Pisarenko and Rodkin [START_REF] Pisarenko | Statistical analysis of natural disasters and related losses[END_REF] and Embrechts et al. [START_REF] Embrechts | Modelling extremal events for insurance and finance[END_REF]. If one observes n independent realizations of X, a quantile Q X (β n ) is said to be large if β n → 0 as n → ∞. When β n does not converge too fast to 0, i.e., nβ n → ∞, the quantile can be simply estimated by an order statistic. A more challenging question arises when nβ n → c > 0 i.e., when Q X (β n ) is an extreme quantile. In this situation a rough knowledge of the shape of the tail distribution is required. In this paper, we assume that F X belongs to a maximum domain of attraction (MDA) that is to say there exist sequences (a n ) > 0 and (b n ) and a real ξ such that

P max(X 1 , • • • , X n ) -b n a n ≤ x = exp -(1 + ξx) -1/ξ ,
for all x satisfying 1 + ξx > 0. This convergence in distribution can be reformulated in terms of the quantile function Q X . More specifically, F X belongs to a MDA if and only if there exist a parameter ξ ∈ R and an auxiliary function a : [1, ∞) → (0, ∞) such that for all t > 0,

lim α→0 Q X (tα) -Q X (α) a(α -1 ) = K ξ (t -1 ) := t -1 1 s ξ-1 ds. (1)
The parameter ξ, called the extreme value index, plays an important role in modeling extreme events since it controls the tail heaviness of the distribution. More precisely, if ξ < 0, the right endpoint Q X (0) is finite and F X decreases at a polynomial rate. When ξ = 0, the survival function decreases at an exponential rate and when ξ > 0, the distribution is said to be heavy-tailed that is to say, for all t > 0,

lim α→0 Q X (tα) Q X (α) = t -ξ .
In other words, the quantile function Q X is heavy-tailed if and only if the function Q X (1/•) is a regularly varying function with index ξ or equivalently,

Q X (1/x) = x -ξ L(x)
where L is a slowly varying function that is to say, for all t > 0, the ratio L(tx)/L(x) converges to 1 as x → ∞.

While the estimation of extreme quantiles has been widely studied in the literature (see for instance the references [START_REF] Weissman | Estimation of parameters and large quantiles based on the k largest observations[END_REF][START_REF] El Methni | Estimation of extreme quantiles from heavy and light tailed distributions[END_REF][START_REF] Daouia | On kernel smoothing for extreme quantile regression[END_REF]), little work has been dedicated to the construction of an asymptotic confidence interval. This is precisely the question addressed in this paper. More specifically, for a preselected level γ ∈ (0, 1), we are searching for positive random variables L n,γ and R n,γ such that

lim n→∞ P (L n,γ ≤ Q X (β n )) = lim n→∞ P (R n,γ ≥ Q X (β n )) = 1 + γ 2 .
The random interval [L n,γ , R n,γ ] is then an asymptotic two-sided confidence interval of level γ for Q X (β n ).

In the recent literature, one can mention the existence of three papers dedicated to the construction of an asymptotic interval for extreme quantiles.

In the first one (Gardes [START_REF] Gardes | Nonparametric confidence interval for conditional quantiles with large-dimensional covariates[END_REF]), the construction is addressed only for quantiles of order β n with nβ n → ∞. The construction of an asymptotic confidence interval for extreme quantiles (nβ n → c > 0) has been tackled in Gardes and Maistre [START_REF] Gardes | Nonparametric asymptotic confidence intervals for extreme quantiles[END_REF] and Buitendag [START_REF] Buitendag | Confidence intervals for extreme Pareto-type quantiles[END_REF]. In both works, the underlying distribution is assumed to be heavy-tailed and the proposed confidence intervals are not suitable when ξ ≤ 0. The novelty brought by this work is the proposition of an asymptotic confidence interval for extreme quantiles that can be used whatever the sign of the extreme value index.

The paper is organized as follows. The proposed confidence interval is introduced in Section 2. Roughly speaking, it is obtained by extrapolating two order statistics over the range of the data. The estimation of these extrapolation factors is considered in Section 3. A general estimation procedure in presented in Section 3.1 where sufficient conditions for the convergence of the coverage probability to the nominal one are given. An application to a classical extreme quantile estimator is given in Section 3.2. The finite sample behavior of our procedure is investigated on simulated data in Section 4 and on a real dataset in Section 5. All the proofs are gathered in Section 6.

Construction of the confidence interval

Let us consider a random variable X such that (1) holds for the quantile function

Q X . Let X 1 , • • • , X n be a sample of n independent replications of X and let X 1,n ≤ • • • ≤ X n,n
be the sample arranged in ascending order. Let us briefly explain the procedure used to find the right bound of the confidence interval, namely the random variable R n,γ . The same procedure can be applied to find the left bound L n,γ . Since the probability space is atomless, there exist independent standard uniform random variables U 1 , • • • , U n such that the random vector (X n-j,n , j = 0, • • • , n -1) is distributed as the random vector (Q X (U j+1,n ), j = 0, • • • , n -1). Hence, for any integers j R ∈ N, it is readily seen that

P (X n-j R ,n ≥ Q X (β n )) = P (U j R +1,n ≤ β n ) = F (j R +1,n-j R ) Beta (β n ) ,
where

F (r,s)
Beta is the cumulative distribution function of a Beta distribution with parameters r > 0 and s > 0. If nβ n → c > 0, we have that

lim n→∞ P (X n-j R ,n ≥ Q X (β n )) = lim n→∞ F (j R +1,n-j R ) Beta (β n ) = F (j R +1,1) Gamma (c) ,
where F (r,s) Gamma is the cumulative distribution function of a Gamma distribution with parameters r > 0 and s > 0, see Lemma 1 for a proof. As a consequence, for a given level β n and a given probability γ, there is no guarantee of finding an integer j R ∈ N such that P

(X n-j R ,n ≥ Q X (β n )) → (1+γ)/2.
As an example, if β n = 1/(2n), we have for all j ∈ N that F (j+1,1) Gamma (1/2) ≤ 0.39 < (1 + γ)/2 for all γ ∈ (0, 1). It thus appears that for a level β n close to 0, an extreme order statistic is not large enough to be considered as a bound of an asymptotic confidence interval for Q X (β n ). We thus need to find a positive extrapolation factor t n,R (depending on j R ) in order to have

P (t n,R X n-j R ,n ≥ Q X (β n )) ≈ (1 + γ)/2.
To find such a sequence, we use the methodology introduced in [START_REF] Gardes | Nonparametric asymptotic confidence intervals for extreme quantiles[END_REF]. More specifically, using the same arguments as above, we get that

P (t n,R X n-j R ,n ≥ Q X (β n )) = F (j R +1,n-j R ) Beta F X (t -1 n,R Q X (β n )) . A good candidate is the sequence t n,R = Q X (β n )/Q X (a n,R (γ)) where, denot- ing by q (r,s) Beta the inverse of F (r,s) beta , a n,R (γ) := q (j R +1,n-j R ) Beta 1 + γ 2 . ( 2
)
Of course, the factor t n,R is unknown and need to be estimated by any consistent estimator t n,R . An example of an estimator is given later. Finally, we propose to take for the right bound of the asymptotic confidence interval for

Q X (β n ), the random variable R n,γ = t n,R X n-j R ,n
. Similarly, we propose to take L n,γ = t n,L X n-j L ,n where t n,L is any consistent estimator of

t n,L := Q X (β n )/Q X (a n,L (γ))
where

a n,L (γ) := q (j L +1,n-j L ) Beta 1 -γ 2 . ( 3
)
To sum up, we give below the definition of the two-sided asymptotic confidence interval for Q X (β n ) that is considered in all what follows.

Definition 1 Let X be a positive random variable such that (1) holds for the quantile function

Q X . Let X 1 , • • • , X n be n independent replications of X.
For a preselected probability γ ∈ (0, 1), a sequence β n such that nβ n → c > 0 and (j L , j R ) ∈ N 2 , the two-sided confidence interval for the extreme quantile

Q X (β n ) is CI(γ, β n ) := t n,L X n-j L ,n , t n,R X n-j R ,n ,
where t n,L and t n,R are any consistent estimators of the ratios t n,L and t n,R with

t n,• := Q X (β n ) Q X (a n,• (γ))
, the symbol • being either L or R.

The confidence interval CI(γ, β n ) is thus a rescaled version of the random interval [X n-j L ,n , X n-j R ,n ]. The choice of the two integers j L and j R is left to the user. A proposition for the couple (j L , j R ) is given in the simulation study.

In the next result, we provide sufficient conditions on the estimators t n,L and t n,R ensuring that the asymptotic coverage probability of CI(γ, β n ) is equal to γ. Before that, let us remind that a sequence of real random variables Z n is O P (1) if for all > 0, there exist N ∈ N and c > 0 such that for all n ≥ N ,

P(Z n ∈ [-c , c ]) > 1 -. In particular, if Z n converges in distribution, then Z n = O P (1).
Theorem 1 Assume that (1) holds for Q X and let

(β n ) ∈ (0, 1) be a se- quence such that nβ n → c > 0. If the extrapolation factors t n,L and t n,R in Definition 1 are such that σ -1 n (t n,• / t n,• -1) = O P (1) where (σ n ) is a positive sequence satisfying lim n→∞ σ n Q X (1/n) a(n) = 0, (4)
then the probabilities

P t n,L X n-j L ,n ≤ Q X (β n ) and P t n,R X n-j R ,n ≥ Q X (β n ) , both converge to (1 + γ)/2 as n → ∞.
From de Haan and Ferreira [11, Lemma 1.2.9], we have

lim n→∞ a(n) Q X (1/n) = max(0, ξ).
Hence, condition (4) entails that the sequence σ n converges to 0 and thus that t n,• is a consistent estimator of t n,• in the sense that the ratio t n,• /t n,• converges in probability to 1. More specifically, if ξ > 0, condition (4) reduces to σ n → 0. When ξ < 0, we have that

Q X (1/n) → Q X (0) =:
x * < ∞ and, according to [11, Lemma 1.2.9 and Corollary 1.

2.10], a(n) ∼ -ξ{x * -Q X (1/n)} = n ξ L(n),
where L is a slowly varying function. So, when ξ < 0, condition (4) takes the form σ n n -ξ /L(n) → 0. In particular, when ξ < -1/2, the ratio t n,• /t n,• must converges to 1 faster than the classical parametric rate of convergence n 1/2 . Constructing an asymptotic confidence interval for Q X (β n ) in this situation seems to be a quite difficult task.

Estimation of the extrapolation factors

Starting from a sample X 1 , • • • , X n of independent copies of X, this section is dedicated to the estimation of the extrapolation factors t n,L and t n,R . Recall that, for some preselected probability γ, these extrapolation factors are given by

t n,• := Q X (β n ) Q X (a n,• (γ))
.

As before, we assume that the quantile function Q X satisfies (1).

General estimation procedure

The natural way to estimate t n,• is to plug any consistent estimator Q X,n of Q X (•) in the expression of t n,• leading to the class of estimators

t ( * ) n,• := Q X,n (β n ) Q X,n (a n,• (γ)) . (5)
For this class of estimators, we have the following version of Theorem 1.

Corollary 1 Assume that (1) holds for Q X and let (β n ) ∈ (0, 1) be a sequence such that nβ n → c > 0. If there exists a positive sequence

(ϑ n ) with ϑ n /a(n) → 0 and ϑ -1 n Q X,n (u n ) -Q X (u n ) = O P (1),
for any sequence (u n ) satisfying nu n → d > 0, then, the conclusion of Theorem 1 holds.

Application: Moment Estimator

A classical estimator of the extreme quantile Q X (u n ) where the sequence

(u n ) ∈ (0, 1) satisfies nu n → c > 0 as n → ∞ is provided by the Moment Estimator Q (M )
X,n (see for instance de Haan and Ferreira [11, Section 4.3.2]) which is based on the moment statistics

M (j) n := 1 k n kn i=1 (ln X n-i+1,n -ln X n-kn,n ) j , j ∈ {1, 2}.
More specifically, for a sequence

(k n ) ∈ {1, • • • , n}, its expression is given by Q (M ) X,n (u n ) := X n-kn,n + a (M ) n n k n K ξ (M ) n k n nu n ,
where,

ξ (M ) n := M (1) n + 1 - 1 2 1 - {M (1) n } 2 M (2) n -1
, is the moment estimator of the extreme value index ξ introduced in Dekkers et al. [START_REF] Dekkers | A moment estimator for the index of an extreme-value distribution[END_REF] and

a (M ) n n k n := X n-kn,n M (1) n 1 -ξ (M ) n + M (1) n ,
is the moment estimator of the auxiliary function a(n/k n ).

To deal with the asymptotic behavior of the Moment Estimator, the rate of convergence in (1) needs to be specified. The following second-order condition is then introduced.

(SOC) There exist ρ < 0, ξ = ρ and functions a > 0 and A with A(x) → 0 as x → ∞ such that for all t > 0,

lim α→0 1 A(α -1 ) Q X (tα) -Q X (α) a(α -1 ) -K ξ t -1 = H ξ,ρ t -1 ,
where

H ξ,ρ (s) := {K ξ+ρ (s) -K ξ (s)}/ρ.
A detailed discussion on this condition can be found in the monographs of The second-order parameter ρ < 0, controls the rate of convergence in (1): the larger the value of ρ, the slower the convergence. Note that when there exists an auxiliary function a such that

Q X (tα) -Q X (α) a(α -1 ) = K ξ (t -1 ),
the second-order parameter ρ is by convention equal to -∞. From [START_REF] De Haan | Extreme Value Theory: An introduction[END_REF]Lemma B.3.16], if Q X satisfies the second-order condition (SOC) then, there exist ρ ∈ [0, ρ] and a function B with B(x) → 0 as x → ∞ such that

lim α→0 1 B(α -1 ) ln Q X (tα) -ln Q X (α) a(α -1 )/Q X (α) -K ξ -t -1 = H ξ -,ρ t -1 , (6)
where ξ -:= min(0, ξ) as before. Note that the function B is regularly varying with index ρ and, as a consequence,

A(x) = O(B(x)).
The asymptotic behavior of Q

(u n ) with nu n → c > 0, {ϑ (M ) n } -1 Q (M ) X,n (u n ) -Q X (u n ) = O P (1),
where

ϑ (M ) n = k -1/2 n a n k n kn 1 s ξ-1 ln(s)ds.
The corresponding asymptotic confidence interval as defined in Definition 1 is given by

CI (M ) (γ, β n ) := t (M ) n,L X n-j L ,n , t (M ) n,R X n-j R ,n , (7)
with

t (M ) n,• = Q (M ) X,n (β n ) Q (M ) X,n (a n,• (γ)) , • ∈ {L, R}.
According to Corollary 1, the random interval

CI (M ) (γ, β n ) is an asymptotic two-sided confidence interval for Q X (β n ) if the condition lim n→∞ ϑ (M ) n a(n) = lim n→∞ a(n/k n ) a(n) 1 k 1/2 n kn 1 s ξ-1 ln(s)ds = 0, (8)
holds. We have the following result.

Corollary 2 Assume that the second-order condition (SOC) holds for Q X and let (β n ) be a sequence such that nβ n → c > 0.

i) If ξ > -1/2, conditions of Proposition 1 hold by taking for (k n ) any sequence asymptotically proportional to n δ with δ = -2(η +ρ )/(1-2ρ ) where η ∈ (0, -ρ ) can be taken as close as we want to 0.

ii) If ξ < -1/2, there does not exist a sequence (k n ) for which condition ( 8) is satisfied.

A clear conclusion cannot be given in the situation where ξ = -1/2. It depends on the asymptotic behavior of the auxiliary function a. When ξ < -1/2, the moment estimator cannot be used to construct a valid confidence interval. As mentioned before, in this case, we need to use an estimator of t n,• that converges faster than the parametric rate n 1/2 . Up to our knowledge, such an estimator is not available in the literature. We conclude this section by a focus on the situation where Q X is heavytailed (i.e., ξ > 0). It is well known that for any intermediate sequence (k n ) (i.e., k n → ∞ and n/k n → ∞ as n → ∞) and any sequence (u n ) with nu n → c > 0, the Weissman estimator [START_REF] Weissman | Estimation of parameters and large quantiles based on the k largest observations[END_REF] given by

Q (W ) X,n (u n ) := X n-kn,n nu n k n -ξn
, where ξ n is the classical Hill estimator [START_REF] Hill | A simple general approach to inference about the tail of a distribution[END_REF], is a consistent estimator of Q X (β n ) (see de Haan and Ferreira [START_REF] De Haan | Extreme Value Theory: An introduction[END_REF]Theorem 4.3.8]). Replacing in (5), Q X,n by the Weissman estimator leads to the asymptotic confidence interval for Q X (β n ) given by

CI (W ) (γ, β n ) := β n a n,L (γ) -ξn X n-j L ,n , β n a n,R (γ) -ξn X n-j R ,n . (9)
This random interval is slightly different from the confidence interval introduced in Gardes and Maistre [START_REF] Gardes | Nonparametric asymptotic confidence intervals for extreme quantiles[END_REF]. The difference lies in the choice of j L and j R . In [START_REF] Gardes | Nonparametric confidence interval for conditional quantiles with large-dimensional covariates[END_REF], j L and j R are assumed to be fixed (independent of the sample size n) while in [START_REF] Gardes | Nonparametric asymptotic confidence intervals for extreme quantiles[END_REF], j L = j R = j n where (j n ) is an intermediate sequence. Of course, the confidence interval ( 9) is only adapted to heavy-tailed quantiles.

Simulation study

In this section, we are interested in the behavior of the confidence interval

CI (M ) (γ, β n ) for Q X (β n
) when a sample of size n ∈ {200; 500; 1,000; 2,000} is observed. For this study, we take γ = .95 and β n = 1/n. Several distributions are considered. The two distributions below correspond to the case ξ > 0 i.e., to heavy-tailed quantiles.

• Absolute student distribution -The quantile function is given for all u ∈ (0, 1) by Q X (u) = q t (1u/2; µ), where q t (•; µ) is the quantile function of a Student distribution with µ > 0 degree of freedom. The extreme-value index is ξ = 1/µ and the second-order parameter is ρ = -1.

• Burr distribution -For κ > 0 and c > 0, the quantile function is given for all u ∈ (0, 1) by Q X (u) = {u -1/κ -1} 1/c . For this distribution, ξ = 1/(cκ) and ρ = -1/κ.

The extreme value index of the next three distributions is ξ = 0. In this case, the distribution is said to be light-tailed.

• Exponential distribution -For all values of the rate parameter λ > 0, the second-order parameter is ρ = -∞.

• Normal distribution -The second-order parameter is ρ = 0 whatever the mean and the standard deviation of the distribution.

• Log-Normal distribution -For µ ∈ R and σ 2 > 0, the quantile function is given for all u ∈ (0, 1) by Q X (u) = exp{q N (u; µ, σ 2 )}, where q N (•; µ, σ 2 ) is the quantile function of a normal distribution with mean µ and variance σ 2 . The second-order parameter is ρ = 0.

Finally, the following distributions belong to a MDA with a negative extreme value index.

• Beta distribution -For α > 0 and β > 0, the probability density function is proportional to x α-1 (1x) β-1 . The extreme value index is ξ = -1/β and the second-order parameter ρ = -∞.

• Reverse Burr distribution -For κ > 0 and c > 0, the quantile function is given for all u ∈ (0, 1) by Q X (u) = 1 -{u -1/κ -1} -1/c . The extreme value index and the second-order parameter are ξ = -1/(cκ) and ρ = -1/κ.

As noticed in the introduction, few works are dedicated to the construction of a confidence interval for an extreme quantile. We can however mention the papers of Buitendag et al. [START_REF] Buitendag | Confidence intervals for extreme Pareto-type quantiles[END_REF] and Gardes and Maistre [START_REF] Gardes | Nonparametric asymptotic confidence intervals for extreme quantiles[END_REF] where this question has been tackled in the case where the underlying distribution is heavy-tailed. As mentioned at the end of Section 3.2, the interval proposed in [START_REF] Gardes | Nonparametric asymptotic confidence intervals for extreme quantiles[END_REF] is a slightly different version of the interval CI (W ) (γ, β n ) given in [START_REF] Gardes | Nonparametric confidence interval for conditional quantiles with large-dimensional covariates[END_REF] which is obtained by using the Weissman estimator instead of the Moment Estimator Q (M ) X,n . Without going further into details, Gardes and Maistre [START_REF] Gardes | Nonparametric asymptotic confidence intervals for extreme quantiles[END_REF] proposed to use the same intermediate sequence (j n ) instead of the integers j L and j R in [START_REF] Gardes | Nonparametric confidence interval for conditional quantiles with large-dimensional covariates[END_REF]. In other words, the left and right bounds in [START_REF] Gardes | Nonparametric asymptotic confidence intervals for extreme quantiles[END_REF] are extrapolations of the intermediate order statistic X n-jn,n while in CI (W ) (γ, β n ), the left (resp. right) bound in an extrapolation of the extreme order statistic X n-j L ,n (resp. X n-j R ,n ). Moreover, Gardes and Maistre [START_REF] Gardes | Nonparametric asymptotic confidence intervals for extreme quantiles[END_REF] introduced a bias-reduced version of their interval (denoted hereafter CI (BR) (γ, β n )) whose performance has been compared to the one of the interval introduced in Buitendag et al. [START_REF] Buitendag | Confidence intervals for extreme Pareto-type quantiles[END_REF]. It appears that the bias-reduced confidence interval in Gardes and Maistre [START_REF] Gardes | Nonparametric asymptotic confidence intervals for extreme quantiles[END_REF] is most of the time associated to a better (or at least equivalent) coverage probability. For the two heavy-tailed distributions in our simulation study, we thus compare the new confidence interval CI (M ) (.95, 1/n) to the bias-reduced interval CI (BR) (.95, 1/n) in [START_REF] Gardes | Nonparametric asymptotic confidence intervals for extreme quantiles[END_REF]. The accuracy of the confidence interval CI (M ) (.95, 1/n) is evaluated by

E (M ) (k n ) := 1 2 |γ L (k n ) -.975| + |γ R (k n ) -.975| , where γ L (k n ) := P t (M ) n,L X n-j L ,n ≤ Q X (1/n) , and γ R (k n ) := P t (M ) n,R X n-j R ,n ≥ Q X (1/n) .
For of CI (M ) (.95, 1/n), a good performance in terms of coverage probability corresponds to a value of E (M ) (k n ) close to 0. Of course, the true value of E (M ) (k n ) is unknown but it can be approximated with a desired precision by means of a Monte Carlo (MC) procedure. In this study, the MC procedure is done by generating N = 25,000 independent samples. When the distribution is heavy-tailed, the same quantity is computed for the bias-reduced interval CI (BR) (.95, 1/n) and it is denoted

E (BR) (k n ).

Selection of the hyper-parameters

First, for a heavy-tailed distribution, the interval CI (BR) (.95, 1/n) depends on the choice of two intermediate sequences: a sequence (k n ) used to compute the Weissman estimator and a sequence (j n ) used in the intermediate order statistic X n-jn,n . The authors suggest to take j n = max{3, ln .85 (k n ) }. The same choice is adopted in this simulation study. The sequence (k n ) is selected by using the procedure described in [START_REF] Gardes | Nonparametric asymptotic confidence intervals for extreme quantiles[END_REF]. The obtained sequence is denoted ( k

(BR) n ).
For the interval CI (M ) (.95, 1/n), which can be used whatever the value of the extreme value index, the choice of two integers j L and j R and of an intermediate sequence (k n ) are required. The integers j L and j R are selected in order to minimize the extrapolation factors t

(M ) n,L and t (M ) n,R in CI (M ) (γ, β n ).
In other words, we take j L and j R so that a n,L (.95) = q (j L +1,n-j L ) Beta (.025) and a n,R (.95)q

(j R +1,n-j R ) Beta
(.975) are as near as possible to β n = 1/n that is to say

j L = arg min j∈N q (j+1,n-j) Beta (.025) - 1 n and j R = arg min j∈N q (j+1,n-j) Beta (.975) - 1 n .
For all n ∈ {200; 500; 1,000; 2,000}, we have j L = 3 and j R = 0.

To select the number of observations k n used for the Moment Estima-tor, we consider a method based on the excesses over a random threshold that has been proposed by Hüsler et al. [START_REF] Hüsler | Extreme value index estimator using maximum likelihood and moment estimation[END_REF]. Let us briefly describe this selection procedure. For a given intermediate sequence

(k n ), let {Z i = X n-i+1,n -X n-kn,n , i = 1, • • • , k n }
H ξ x f (t) := 1 -1 + ξ x f (t) -1/ξ for all x > 0 such that 1 + ξx/f (t) > 0, where f (t) = a(1/(1 -F X (t))).
Hence, since the intermediate order statistic X n-kn,n is a consistent estimator of Q X (k n /n) we can reasonably assume that, for n large enough and for a well chosen sequence (k n ), the random variables Z 1 , • • • , Z kn are independent with common distribution function H ξ (•/a(n/k n )) and consequently that the random vector

(H ξ (Z 1 /a(n/k n )), • • • , H ξ (Z kn /a(n/k n ))) is distributed
D(k n ) := 1 k n kn i=1 H ξ (M ) n Z i a (H) n (n/k n ) - k n -i + 1 k n + 1 .
In view of the previous discussion, if (k n ) is well chosen, the discrepancy measure D(k n ) should be close to 0. The selected value of k n is thus given by k (M )

n := arg min k∈{k 1 ,••• ,k 2 } D(k),
where

{k 1 , • • • , k 2 } ⊂ {1, • • • , n}.
In this simulation study, we take k 1 = .02n and k 2 = .8n . The results are gathered in Tables 1 to 5. Let us first emphasize that for all the considered distributions and for all sample sizes, the choice

k n = k (M ) n
for the confidence interval CI (M ) (.95, 1/n) leads to better results in terms of coverage probability, than the arbitrary choice k n = n/5 . Note also that, as expected, the accuracy measure

E (M ) ( k (M )
n ) decreases as the sample size increases. Next, even if CI (M ) (.95, 1/n) is not specifically dedicated for heavy-tailed distributions (while CI (BR) (.95, 1/n) is), its accuracy is better for the student distribution with µ = 2 and the Burr distribution as soon as n ≥ 500.

Real data study

Jeanne Calment was a French supercentenarian (i.e., a person who has reached the age of 110 years) who was born on 21 February 1875 and died on 4 August 1997 at the age of 122 years and 164 days. This exceptional longevity makes her the longest-living human at least among those whose birth and death dates could be verified. To get an idea of this exceptional lifespan, the second longest human lifespan observed in France (and in Europe) was 118 years and 341 days. We would like to check whether Jeanne Calment's lifespan is an outlier or not. For this purpose, we compare her with French supercentenarians born during a similar period. Our study is based on the International Database on Longevity (IDL) which is the product of an international collaboration of several statistical offices and a large number of experts. This database contains birth and death dates of supercentenarians for 13 countries, and semi-supercentenarians (persons with a lifespan of between 105 and 110 years old) for 10 countries. The population chosen for our comparison is the French supercentenarians and semi-supercentenarians born between 1890 and 1899. The number of observations per year over this period is comparable ranging from 162 for the year 1890 to 310 for the year 1898. We denote by x 1 , • • • , x n the lifespans in years of the n = 2400 individuals in the study. We assume that these observations are realizations of n independent replications X 1 , • • • , X n of a random variable X. In terms of the years studied and the period covered by the IDL , none of the persons concerned is still alive. This allows us to state that the variable X is not right-censored because of omissions of supercentenarians not included in the sample due to their longevity. We can consider that the lifespan of Jeanne Calment is not an outlier if it is close to the 1/2400-quantile which is the lifespan that is expected to be observed every 2400 people over the age of 105. We thus check if Jeanne Calment's lifespan belongs to the observed 95% confidence interval for Q X (1/2400). We proceed with the confidence interval based on the Moment Estimator (see Section 3.2). The different parameters are taken following the procedure described in section 4. The obtained parameters are j = 3, j R = 0 and k (M ) n = 1388. The extreme value index is estimated by ξ X = -0.059 and the observed confidence interval is [113.8; 116.1]. Clearly, the value 122.5 is much larger than the upper bound. It thus seems that Jeanne Calment's lifespan can be considered as an outlier. To go further, for each year, the 95% confidence intervals for the 1/250-th quantile are represented in Figure 1. As expected, the lifespan of Jeanne Calment is clearly larger the all the upper bounds. We also note that the estimates of the extreme value index are always close to 0. This is in line with several studies such as the one of Alvarez et al. [START_REF] Alvarez | Regularities in human mortality after age 105[END_REF] where it is shown that the risk of dying appears constant over the age of 105. This surprising phenomenon is known as the "mortality plateau". In other words, it seems that X follows an exponential law (constant hazard rate).

Proofs

Preliminary results

Lemma 1 For all integer j and for all δ ∈ (0, 1),

lim n→∞ nq (j+1,n-j) Beta (δ) = q (j+1,1) Gam (δ),
where q (r,s) Gam is the quantile function of a Gamma distribution with parameters r > 0 and s > 0.

Proof -Let us first give some notations. Let E 1 , • • • be independent standard exponential random variables and, for k ∈ N\{0}, let

T k := E 1 +• • •+E k .
The distribution of T k is then a Gamma distribution with parameters k and 1. and the proof is complete.

From [START_REF] De Haan | Extreme Value Theory: An introduction[END_REF]Theorem B.2.21], condition (1) on the quantile function Q X is equivalent to

lim n→∞ F X (x n + tg(x n )) F X (x n ) = 1 K ← ξ (t) , (10)
where (x n ) is a sequence converging to the right endpoint Q X (0) and g(x) := a(1/F X (x)). A generalization of (10) when t is replaced by a sequence t n → t 0 was proved in [5, Lemma 2].

Lemma 3 Assume that (1) holds and let (x n ) be a sequence converging to the right endpoint Q X (0). For all sequence (t n ) such that t n → t 0 , if there exists η > 0 for which 1 + ξt 0 ≥ η then,

lim n→∞ F X (x n + t n g(x n )) F X (x n ) = 1 K ← ξ (t 0 ) .
6.2 Proofs of main results

Proof of Theorem 1

We need to prove that

P t n,L X n-j L ,n ≤ Q X (β n ) → 1 + γ 2 , (11)
and

P t n,R X n-j R ,n ≥ Q X (β n ) → 1 + γ 2 . ( 12
)
We only prove [START_REF] De Haan | Extreme Value Theory: An introduction[END_REF], the proof of (12) being similar. Let > 0. By assumption, there exist c > 0 and N ∈ N such that for all n ≥ N ,

P A n, := σ -1 n t n,L t n,L -1 ∈ [-c , c ] ≥ 1 -,
where we recall that t n,L := Q X (β n )/Q X (a n,L (γ)). We have the decomposition

P t n,L X n-j L ,n ≤ Q X (β n ) = P t n,L X n-j L ,n ≤ Q X (β n ) ∩ A n, + P t n,L X n-j L ,n ≤ Q X (β n ) ∩ Ω \ A n, =: T 1,n + T 2,n . Obviously, 0 ≤ T 2,n ≤ , (13)
and thus T 2,n → 0 as n → ∞. Let us focus on the term T 1,n . We have

T 1,n = P F X (X n-j L ,n ) ≥ F X Q X (β n ) t n,L ∩ A n, . Let x n,L := Q X (a n,L (γ)
). When the event A n, occurs, we have for all n ≥ N ,

F X (x n,L + s n, g (x n,L )) ≤ F X Q X (β n ) t n,L ≤ F X (x n,L -s n, g (x n,L )) ,
where g(y) = a(1/F X (y)) and s n, := c σ n x n,L /g(x n,L ). Lemma 1 entails that a n,L (γ) is asymptotically proportional to 1/n. Using the fact that Q X and a are regularly varying functions, respectively with indices max(0, ξ) and ξ, we get that s n, is asymptotically proportional to Q X (1/n)/a(n). Hence, by assumption, s n, → 0 as n → ∞. Furthermore, x n,L converges to the right endpoint Q X (0) since a n,L (γ) converges to 0. We can thus apply Lemma 3 leading to

F X (x n,L ± s n, g (x n,L )) ∼ F X (x n,L ) = a n,L (γ).
As a consequence, there exists N ( * ) ∈ N such that for all n ≥ max(N , N

* ) ), a n,L (γ)(1 -) ≤ F X Q X (β n ) t n,L ≤ a n,L (γ)(1 + ). (
Note that F X (X n-j L ,n ) is distributed as a Beta distribution with parameters j L + 1 and nj L . We obtain for n ≥ max(N , N ( * ) ),

T 1,n ≤ P F X (X n-j L ,n ) ≥ a n,L (γ)(1 -) ∩ A n, ≤ P F X (X n-j L ,n ) ≥ a n,L (γ)(1 -) = F (j L +1,n-j L ) Beta (a n,L (γ)(1 -)) .
where Z n = O P (1) and Z * n = O P (1). Let ξ + := max(0, ξ). From [11, Lemma 1.2.9 and Corollary 1.2.10], we have as n

→ ∞ that Q X (β n ) ∼ c -ξ + Q X (1/n) and Q X (a n,• (γ)) ∼ {c * } -ξ + Q X (1/n). Moreover, since a(n)/Q X (1/n) → ξ + , it is readily seen that assumption ϑ n /a(n) → 0 entails that ϑ n /Q X (1/n) → 0.
We deduce from these results that

t ( * ) n,• t n,• -1 = O P ϑ n Q X (1/n) .
An aplication of Theorem 1 with σ n = ϑ n /Q X (1/n) concludes the proof.

Proof of Corollary 2

Proof of part i): First, assume that ξ > 0. From [11, Remark 4.3.3] and since

k n → ∞, we have as n → ∞ kn 1 s ξ-1 ln(s)ds ∼ 1 ξ k ξ n ln(k n ).
Moreover, the auxiliary function a is regularly varying with index ξ > 0 (see [11, Lemma 1.2.9 and Corollary 1.2.10]). Hence, there exists a slowly varying function L such that

a(n/k n ) a(n) = k -ξ n L(n/k n ) L(n) .
We then get that as n → ∞,

ϑ (H) n a(n) ∼ 1 ξ k -1/2 n L(n/k n ) L(n) ln(k n ).
Taking k n asymptotically equivalent to n δ with δ > 0 is sufficient to ensure that ϑ

(H) n /a(n) → 0 and it is readily seen that conditions k n → ∞, n/k n → ∞ and k 1/2 n B(n/k n ) → λ ∈ R are checked if δ = -2(η + ρ )/(1 -2ρ )
where η ∈ (0, -ρ ) can be taken as close as we want to 0. Assume now that ξ = 0. We have from [START_REF] De Haan | Extreme Value Theory: An introduction[END_REF]Remark 4.3.3]

that, as n → ∞, kn 1 s -1 ln(s)ds ∼ 1 2 ln 2 (k n ).
Since a is in this case a slowly varying function, it is easy to check that ϑ (H) n /a(n) → 0 for any sequence (k n ) which is asymptotically equivalent to n δ with δ > 0. We next conclude as in the case ξ > 0.

Finally, if ξ ∈ (-1/2, 0), we have from [START_REF] De Haan | Extreme Value Theory: An introduction[END_REF]Remark 4.3.3] that, and thus, as n → ∞,

ϑ (M ) n a(n) ∼ 1 ξ 2 k -(ξ+1/2) n a(n/k n ) a(n) . ( 15
)
Since a is a slowly varying function, we conclude as in the case ξ = 0.

Proof of part ii): We start with [START_REF] Thompson | On confidence ranges for the median and other expectation distributions for populations of unknown distribution form[END_REF]. It appears that

ϑ (M ) n a(n) ∼ U (n/k n ) U (n) ,
where U is a regularly varying function with index ξ + 1/2 < 0. Using [3, Theorem 1.

E (M ) ( k (M ) n ) 2.

Proposition 1

 1 (u n ) has been established in de Haan and Ferreira [11, Section 4.3.2]. Let us recall the obtained result. Assume that the second-order condition (SOC) holds. For any intermediate sequence (k n ) satisfying k 1/2 n B(n/k n ) → λ ∈ R where B satisfies (6), we have, for any sequence

n

  as the order statistics (U kn,kn , • • • , U 1,kn ) of a standard uniform distribution. Replacing ξ and a(n/k n ) by their moment estimators ξ (n/k n ) yields to the discrepancy measure defined by

5 . 2 ]

 52 , since k n → ∞ and n/k n → ∞, we have for any a > 0 and for n large enough,U (n) U (n/k n ) k ξ+1/2 n ≤ sup λ≥a U (λn/k n ) U (n/k n ) λ ξ+1/2 → 0.Hence, since kξ+1/2 n → 0, we have U (n)/U (n/k n ) → 0 proving that, for any sequence (k n ) satisfying k n → ∞ and n/k n → ∞, ϑ (M ) n /a(n) → ∞.n = 200 n = 500 n = 1000 n = 2000

  Beirlant et al. [2, Section 3.3] and de Haan and Ferreira [11, Section 2.3].

  be the k n excesses over the random threshold X n-kn,n . From[START_REF] De Haan | Extreme Value Theory: An introduction[END_REF] Lemma 3.4.1], given X n-kn = t, the random vector (Z 1 , • • • , Z kn ) is distributed as the random vector of order statistics (Z kn,kn , • • • , Z 1,kn ) obtained from independent and identically distributed random variables Z 1 , • • • , Z kn . Moreover, under (1), for t large enough, the common distribution of the Z i can be approximated by a generalized Pareto (GP) distribution with cumulative distribution function given by

Table 1 :

 1 046 % 1.45 % 0.970 % 0.614 %E (BR) ( k E (M ) ( n/5 ) 3.274 % 2.194 % 1.398 % 0.830 % E (BR) ( n/5 ) 0.350 % 0.954 % 1.912 % 1.828 % Student distribution with µ = 2. Value of the accuracy measuresE (M ) (k n ) and E (BR) (k n ) for different choice of the intermediate sequence (k n ). n = 200 n = 500 n = 1000 n = 2000 E (M ) ( k

	(BR) n	) 0.447 % 1.747 % 2.338 % 2.835 %
	(M ) n ) 1.526 % 0.932 % 0.622 % 0.292 %
	E (BR) ( k n (BR)	) 1.420 % 0.585 % 0.261 % 0.482 %
	E (M ) ( n/5 ) 2.948 % 1.732 % 1.020 %	0.52 %
	E (BR) ( n/5 ) 2.31 % 0.970 % 0.709 % 0.268 %

Table 2 :

 2 Student distribution with µ = 1. Value of the accuracy measuresE (M ) (k n ) and E (BR) (k n ) for different choice of the intermediate sequence (k n ).

	n = 200 n = 500 n = 1000 n = 2000
	Exponential distribution with λ = 1/2
	E (M ) ( k n ) 1.206 % 0.892 % 0.452 % 0.146 % (M )
	(M ) n ) 0.886 % 0.418 % 0.334 % 0.582 %
	E (M ) ( n/5 ) 2.316 % 1.174 % 0.284 % 0.486 %

E (M ) ( n/5 ) 2.106 % 1.310 % 0.500 % 0.146 % Normal distribution with µ = 0 and σ 2 = 1

E (M ) ( k (M ) n ) 1.776 % 1.782 % 1.410 % 1.260 % E (M ) ( n/5 ) 2.

334 % 2.116 % 1.898 % 1.812 % Log-Normal distribution with µ = 0 and σ 2 = 1 E (M ) ( k

Table 4 :

 4 Gumbel MDA. Value of the accuracy measure E (M ) (k n ) for different choice of the intermediate sequence (k n ). n = 200 n = 500 n = 1000 n = 2000 Beta distribution with α = 4 and β = 4 E (M ) ( k

	(M )
	(M ) n ) 0.736 % 0.848 % 0.752 % 0.596 %
	E (M ) ( n/5 ) 2.082 % 2.508 % 2.886 % 3.112 %

n ) 2.136 % 2.168 % 2.174 % 1.930 % E (M ) ( n/5 ) 2.442 % 2.256 % 2.226 % 2.206 % Reverse Burr distribution with κ = 1 and c = 1/4 E (M ) ( k (M ) n ) 1.478 % 1.462 % 1.138 % 0.890 % E (M ) ( n/5 ) 2.190 % 2.078 % 1.918 % 1.774 % Reverse Burr distribution with κ = 1 and c = 1/2 E (M ) ( k

Table 5 :

 5 Weibull MDA. Value of the accuracy measure E (M ) (k n ) for different choice of the intermediate sequence (k n ).

From Rényi's representation of ordered standard uniform random variables, the random vector (U j+1,n , j = 0, • • • , n -1) is distributed as the random vector (T j+1 /T n+1 , j = 0, • • • , n -1). Since U j+1,n is distributed according to a Beta distribution with parameters j + 1 and nj, we have

Moreover, from the law of large numbers, n/T n+1 converges almost surely to 1 and thus nT j+1 /T n+1 converges in distribution to a Gamma distribution with parameters k and 1. As a consequence, since the cumulative distribution function F (j+1,1) Gam of a Gamma distribution is continuous, we have

as n → ∞. We conclude the proof by using the continuity of q (j+1,n-j) Beta

.

Beta be the density function of a Beta distribution with parameters r > 0 and s > 0. For any sequence (z n ) ∈ (0, 1) such that nz n → c > 0 as n → ∞ and for any positive integer j, we have

Proof -The density of a Beta distribution with parameters j + 1 and nj is given by

Stirling's formula ensures that, as n → ∞, we have n! ∼ √ 2πn n+1/2 e -n . Replacing in the above expression n! and (nj -1)! by their equivalents leads to n! n j (nj -1)! ∼ n, Moreover, still for n ≥ max(N , N ( * ) ),

To sum up, we have proved that for all > 0 and for all n ≥ max(N , N ( * ) ),

To prove that T 1,n → 0 and to conclude the proof, it remains to show that

Recall that F

The mean value theorem entails that there exists κ ∈ (0, 1) such that

From Lemma 1, there exists c 1 = c 1 (γ, j L ) > 0 such that a n,L (γ) ∼ c 1 /n. Hence, from Lemma 2

It is then readily seen that there exists positive constants c 3 and c 4 such that for all > 0 and for n large enough,

.

Proof of Corollary 1

By assumption, nβ n → c > 0 and, from Lemma 1, there exists c * > 0 such that na n,• (γ) → c * . Hence, using the assumption on the asymptotic behavior of the extreme quantile estimator Q X,n , we get