On the Generalized Hill Process for Small Parameters and Applications

2011/11/19 by Gane Samb Lô, Lo, Gane Samb, El Hadji Dème +3
Computer Science · Economics, Econometrics and Finance · Mathematics · #60F05. Secondary : 60B10 #60F17 #62F12 #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #Financial Risk and Volatility Modeling #Methodology (stat.ME) #Primary : 62E20 #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1111.4564

openalex publication_date 2011/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X₁,X₂,... be a sequence of independent copies (s.i.c) of a real random variable (r.v.) X≥ 1, with distribution function df F(x)=ℙ% (X≤ x) and let X_1,n≤ X_2,n ≤ ... ≤ X_n,n be the order statistics based on the n≥ 1 first of these observations. The following continuous generalized Hill process equation* T_n(τ)=k^-τ∑_j=1^j=kj^τ(log X_n-j+1,n-log X_n-j,n), equation* τ>0, 1≤ k ≤ n, has been introduced as a continuous family of estimators of the extreme value index, and largely studied for statistical purposes with asymptotic normality results restricted to τ> 1/2. We extend those results to 0 < τ≤ 1/2 and show that asymptotic normality is still valid for τ=1/2. For 0 < τ<1/2, we get non Gaussian asymptotic laws which are closely related to the Riemann function % ζ(s)=∑_n=1^∞ n^-s,s>1

Related

• A double-indexed functional Hill process and applications
• On a discrete Hill's statistical process based on sum-product statistics and its finite-dimensional asymptotic theory
• A functional Generalized Hill process and applications
• Limit laws for the norms of extremal samples
• A supermartingale argument for characterizing the Functional Hill\n process weak law for small parameters