Asymptotic distributions for weighted power sums of extreme values
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Let X_1,n≤⋯≤ 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 γ. Given known constants d_i,n, 1≤ i≤ n, consider the weighted power sums ∑^k_n_i=1d_n+1-i,nlog^pX_n+1-i,n, where p>0 and the k_n are positive integers such that k_n→∞ and k_n/n→0 as n→∞. 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 γ.
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