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.) $X\ge 1$, with distribution function $df$ $F(x)=\mathbb{P}$ and let $X_{1,n}\le X_{2,n}\le ...\le X_{n,n}$ be the order statistics based on the $n\ge 1$ first of these observations. The following continuous generalized Hill process {equation*} T_{n}(τ)=k^{-τ}\sum_{j=1}^{j=k}j^τ(\log X_{n-j+1,n}-\log X_{n-j,n}), \label{dl02} {equation*} $\tau >0$, $1\le k\le n$, 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 $0<\tau \le 1/2$ 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

19 pages; 4 figures

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