On the Generalized Hill Process for Small Parameters and Applications

Details On the Generalized Hill Process for Small Parameters and Applications On the Generalized Hill Process for Small Parameters and Applications

2011-11-19

arxiv.org/abs/1111.4564v1

Statistics

Methodology

19 pages; 4 figures

Scientific paper

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 $% \zeta(s)=\sum_{n=1}^{\infty} n^{-s},s>1$

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