On the Generalized Hill Process for Small Parameters and Applications

Abstract

Let $X_{1},X_{2},...$ be a sequence of independent copies (s.i.c) of a real random variable (r.v.) $X\geq 1$, with distribution function $df$ $F(x)=\mathbb{P}% (X\leq x)$ and let $X_{1,n}\leq X_{2,n} \leq ... \leq X_{n,n}$ be the order statistics based on the $n\geq 1$ first of these observations. The following continuous generalized Hill process {equation*} T_{n}(\tau)=k^{-\tau}\sum_{j=1}^{j=k}j^{\tau}(\log X_{n-j+1,n}-\log X_{n-j,n}), \label{dl02} {equation*} $\tau >0$, $1\leq k \leq n$, has been introduced as a continuous family of estimators of the extreme value index, and largely studied for statistical purposes with asymptotic normality results restricted to $\tau > 1/2$. We extend those results to $0 < \tau \leq 1/2$ and show that asymptotic normality is still valid for $\tau=1/2$. For $0 < \tau <1/2$, we get non Gaussian asymptotic laws which are closely related to the Riemann function $% \zeta(s)=\sum_{n=1}^{\infty} n^{-s},s>1$