A general estimator for the extreme value index: applications to conditional and heteroscedastic extremes

Abstract

The tail behavior of a survival function is controlled by the extreme value index. The aim of this paper is to propose a general procedure for the estimation of this parameter in the case where the observations are not necessarily distributed from the same distribution. The idea is to estimate in a consistent way the survival function and to apply a general functional to obtain a consistent estimator for the extreme value index. This procedure permits to deal with a large set of models such as conditional extremes and heteroscedastic extremes. The consistency of the obtained estimator is established under general conditions. A simulation study and a concrete application on financial data are proposed to illustrate the finite sample behavior of the proposed procedure.

Tail dimension reduction for extreme quantile estimation

“…Since τ −1 n ERV(α n , u|X = x) → 0 locally uniformly, one can use [16,Lemma 3] entailing that for u ∈ [ν, 1] and u ∈…”

“…Note that this estimator belongs to the class of estimators introduced in Gardes [16]. Concerning the estimation of a(α −1 n |x), we consider the statistic…”

“…The first one is a classical regularity condition of the function P(Y > y|B 0 X = •) for large values of y. This condition is essential in a conditional framework (see for instance Daouia et al [7] and Gardes [16]). A careful reading of the proof shows that this condition only needs to be satisfied for x ∈ A.…”

“…. , n. We thus choose to use only a random subsample of size n 0 := an for a given a ∈ (0, 1) to compute (16). Extensive simulation studies show that this procedure does not affect too much the quality of the obtained estimator of B 0 (see Section 4.3).…”

“…According to the third remark after Theorem 2, this choice ensures that estimatorsQ n (β n |B 0 , x) andQ n (β n |x) share the same rate of convergence. To compute the conditional tail index estimator (7) and the auxiliary function estimator (8) we take ν = 0.02 and ϕ(•) = ln(1/•) (this choice was suggested in [16]). Concerning the kernel function K(•) we take the Epanechnikov kernel.…”