Limit laws for the norms of extremal samples
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Let denote S_n(p) = k_n^-1∑_i=1^k_n( log (X_n+1-i,n / X_n-k_n, n) )^p, where p > 0, k_n ≤ n is a sequence of integers such that k_n →∞ and k_n / n → 0, and X_1,n≤...≤ X_n,n is the order statistics of iid random variables with regularly varying upper tail. The estimator γ(n) = (S_n(p)/Γ(p+1))^1/p is an extension of the Hill estimator. We investigate the asymptotic properties of S_n(p) and γ(n) both for fixed p > 0 and for p = p_n →∞. We prove strong consistency and asymptotic normality under appropriate assumptions. Applied to real data we find that for larger p the estimator is less sensitive to the change in k_n than the Hill estimator.
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