Limit laws for the norms of extremal samples

2020/04/27 by Kevei, Peter, Oluoch, Lillian, Viharos, Laszlo
#60F05 #62G32 #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2004.12736

Abstract

Let denote S_n(p) = k_n^-1 ∑_i=1^k_n ( log (X_n+1-i,n / X_{n-kn, 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 \widehat 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 \widehat 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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