How many statistics are needed to characterize the univariate extremes

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.