Estimating the mean of heavy-tailed distributions
type: Item
instance of: scholarly article
title: Estimating the mean of heavy-tailed distributions
zbMATH Open document ID: 1063.62073
author: Joachim Johansson
published in: Extremes
publication date: 2005-05-20T00:00:00Z
review text: For i.i.d. observations \(X=(X_1,\dots,X_n)\) with CDF \[ F(x)=1-cx^{-1/\xi}(1+x^{-\delta}L(x)) \] (\(L\) being a slowly varying function) the problem of mean \({\mathbf E}X_1\) estimation is considered in the case \(\xi\in(1/2,1)\). (For \(\xi\in (0,1/2)\) the sample mean is an asymptotically normal estimate of \({\mathbf E}X_1\), for \(\xi>1\) the mean doesn't exist). The author considers an estimate \[ \widehat E=\int_0^{u_n}x\,dF_n(x)+\int_{u_n}^\infty x\,d\widehat F(x), \] where \(F_n(x)\) is the empirical CDF evaluated by \(X\) and \(\widehat F\) is an estimate of the tail of \(F\). The tail estimation based on the Pareto approximation leads to the estimator \[ \widehat M= n^{-1} \sum_{i=1}^nX_i{\mathbf 1}_{\{X_i\leq u\}} +\widehat p_n(u_n+\widehat\beta_n/(1-\widehat\xi_n)), \] where \(\widehat p=1-F_n(u_n)\), \(\widehat\xi_n\) is an estimate of \(\xi\) and \(\widehat\beta_n\) is an estimate of \(u_n\xi\) (possibly different from \(u_n\widehat\xi_n\)). The asymptotic normality of \(\widehat M\) is demonstrated. Results of simulations are presented. Applications to telecommunication data are considered.
Mathematics Subject Classification ID: 62G32
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62F10
Mathematics Subject Classification ID: 62P30
zbMATH DE Number: 2169511
zbMATH Keywords: Pareto distribution
zbMATH Keywords: Pareto approximation
zbMATH Keywords: asymptotic normality
MaRDI profile type: Publication
full work available at URL: b:extr.0000025668.95782.3d
OpenAlex ID: W2066831115
DOI: 10.1023/B:EXTR.0000025668.95782.3D
description: scientific article; zbMATH DE number 2169511