How many statistics are needed to characterize the univariate extremes

TL;DR

This paper develops a set of nine statistical measures that can determine whether a distribution belongs to the extremal domain of attraction, advancing the understanding of univariate extreme value characterization.

Contribution

It introduces a nine-element empirical characterizing statistics family for univariate extremes and explores the possibility of reducing its size.

Findings

Constructed a nine-element ECSFEXT for extremal domain characterization.

Characterized subdomains within the extremal domain.

Initiated the investigation into minimizing the number of statistics needed.

Abstract

Let $X_{1},X_{2},...$ be a sequence of independent random variables ($rv$) with common distribution function ($df$) $F$ such that $F(1)=0$. We consider the simple statistical problem : find a statistics family of size $m\geq 1$ whose convergence, in probability or almost surely, to a point of some domain $\mathcal{S} \in \mathbb{R}^{m}$ is equivalent that $F$ lies in the extremal domain of attraction $\Gamma$. Such a family, whenever it exists, is called an Empirical Characterizing Statistics Family for the EXTtremes (ECSFEXT). The departure point of this theory goes back to Mason, who proved that the Hill estimator converges a.s. to a positive real number for some particular sequences if and only $F$ lies in the attaction domain of a Fr\'echet's law. Considered for the whole attraction domain, the question becomes more complex. We provide here an ECSFEXT of nine (9) elements and also characterize the subdomains of $\Gamma$. The question of lowering m=9 to a minimum number is launched.