A double-indexed functional Hill process and applications

TL;DR

This paper studies the asymptotic behavior of a double-indexed functional Hill process, introduces new estimators for the extreme value index, and compares their performance with existing methods.

Contribution

It provides the finite-dimensional asymptotic law of a new class of estimators based on a double-indexed process for extreme value analysis.

Findings

Derived the asymptotic distribution of the process's margins.

Introduced new estimators for the extreme value index.

Compared performance of new estimators with existing ones.

Abstract

Let $X_{1,n} \leq .... \leq X_{n,n}$ be the order statistics associated with a sample $X_{1}, ...., X_{n}$ whose pertaining distribution function (% \textit{df}) is $F$. We are concerned with the functional asymptotic behaviour of the sequence of stochastic processes \begin{equation} T_{n}(f,s)=\sum_{j=1}^{j=k}f(j)\left(\log X_{n-j+1,n}-\log X_{n-j,n}\right)^{s}, \label{fme} \end{equation} indexed by some classes $\mathcal{F}$ of functions $f:\mathbb{N}% ^{\ast}\longmapsto \mathbb{R}_{+}$ and $s \in ]0,+\infty[$ and where $k=k(n)$ satisfies \begin{equation*} 1\leq k\leq n,k/n\rightarrow 0\text{as}n\rightarrow \infty . \end{equation*} \noindent We show that this is a stochastic process whose margins generate estimators of the extreme value index when $F$ is in the extreme domain of attraction. We focus in this paper on its finite-dimension asymptotic law and provide a class of new estimators of the extreme value index whose performances are compared to analogous ones. The results are next particularized for one explicit class $\mathcal{F}$.