On the Generalized Hill Process for Small Parameters and Applications

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 $df$ $F(x)=\mathbb{P}$ 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}(τ)=k^{-τ}\sum_{j=1}^{j=k}j^τ(\log X_{n-j+1,n}-\log X_{n-j,n}), \label{dl02} {equation*} $τ>0$, $1\leq k \leq n$, 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 $τ> 1/2$. We extend those results to $0 < τ\leq 1/2$ and show that asymptotic normality is still valid for $τ=1/2$. For $0 < τ<1/2$, we get non Gaussian asymptotic laws which are closely related to the Riemann function $$

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