Asymptotic distributions for weighted power sums of extreme values

Abstract

Let $X_{1,n}\le \cdots \le X_{n,n}$ be the order statistics of $n$ independent random variables with a common distribution function $F$ having right heavy tail with tail index $\gamma$. Given known constants $d_{i,n}$, $1\le i\le n$, consider the weighted power sums ${\sum }_{i=1}^{k_n}{d}_{n+1-i,n}{\log }^p{X}_{n+1-i,n}$, where $p>0$ and the $k_n$ are positive integers such that $k_n\to \infty$ and $k_n/n\to 0$ as $n\to \infty$. Under some constraints on the weights $d_{i,n}$, we prove asymptotic normality for the power sums over the whole heavy-tail model. We apply the obtained result to construct a new class of estimators for the parameter $\gamma$.