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 $\\sum^{k_n}_{i=1}d_{n+1-i,n}\\log^pX_{n+1-i,n}$, where $p>0$ and the $k_n$ are positive integers such that $k_n\\to\\infty$ and $k_n/n\\to0$ 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 $\\gamma$.