A double-indexed functional Hill process and applications

2016/04/16 by Ngom, Modou, Lo, Gane Samb
#60F05. Secondary 62F12 #62G20 #FOS: Computer and information sciences #Methodology (stat.ME) #Primary 62EG32

paper · doi:10.48550/arxiv.1604.04793

Abstract

Let X_1,n ≤ .... ≤ X_n,n be the order statistics associated with a sample X₁, ...., X_n whose pertaining distribution function (% df) is F. We are concerned with the functional asymptotic behaviour of the sequence of stochastic processes T_n(f,s)=∑_j=1^j=kf(j)(log X_n-j+1,n-log X_n-j,n)^s, indexed by some classes F of functions f:ℕ% ^∗\longmapsto ℝ₊ and s ∈ ]0,+∞[ and where k=k(n) satisfies 1≤ k≤ n,k/n→ 0asn→ ∞ . \noindent We show that this is a stochastic process whose margins generate estimators of the extreme value index when F is in the extreme domain of attraction. We focus in this paper on its finite-dimension asymptotic law and provide a class of new estimators of the extreme value index whose performances are compared to analogous ones. The results are next particularized for one explicit class F.

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