Asymptotic distributions for weighted power sums of extreme values - Oluoch Lillian Achola; Viharos László (2021)
Let X1,n ≤ · · · ≤ Xn,n be the order statistics of n independent random variables with a common distribution function F having right heavy tail with tail index γ. Given known constants di,n, 1 ≤ i ≤ n, consider the weighted power sums Pkn i=1 dn+1−i,n logp Xn+1−i,n, where p > 0 and the kn are positive integers such that kn → ∞ and kn/n → 0 as n → ∞. Under some constraints on the weights di,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 γ.