ACTA issues

Asymptotic distributions for weighted power sums of extreme values

Lillian Achola Oluoch, László Viharos

Acta Sci. Math. (Szeged) 87:1-2(2021), 331-346

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 $k_n/n\to 0$ 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 $.

DOI: 10.14232/actasm-020-323-9

AMS Subject Classification (1991): 60F05, 62G32

Keyword(s): tail index, regular variation, weighted power sum, maximum domain of attraction

received 3.7.2020, revised 28.1.2021, accepted 29.1.2021. (Registered under 73/2020.)