Tail behavior, modes and other characteristics of stable distribution
type: Item
instance of: scholarly article
title: Tail behavior, modes and other characteristics of stable distribution
zbMATH Open document ID: 0947.60016
DOI: 10.1023/A:1009908026279
author: Hippolite Fofack
author: John P. Nolan
published in: Extremes
publication date: 2000-05-24T00:00:00Z
review text: A family of stable distributions is considered, i.e. distributions with the characteristic functions \(E\exp(i\theta X)= \exp(-|\theta|^\alpha[1+ i\beta\tan(\alpha\pi/2)\text{sign}(\theta)|\theta|^{1-\alpha}-1] \) if \(\alpha\not=1\) and \(E\exp(i\theta X)= \exp(-|\theta|[1+ i\beta(2/\pi)\text{sign}(\theta)\ln|\theta|]) \) if \(\alpha=1\). Denote the corresponding density by \(f(x;\alpha,\beta)\). It is well known that \(f(x;\alpha,\beta)\sim f_{\text{Pareto}}(x;\alpha,\beta)= \alpha(1+\beta)C_\alpha x^{-(1+\alpha)}\) as \(x\to\infty\). The authors analyze numerically the rate of convergence of \(L_{\alpha,\beta}(x)=f(x;\alpha,\beta)/ f_{\text{Pareto}}(x;\alpha,\beta)\to 1\) and analogous characteristics. The plots presented in the article demonstrate quite unusual behavior of these characteristics with abrupt changes. The authors show how this specific behavior of the tails implies the incorrectness of the Hill estimator for the Pareto index \(\alpha\) of a stable distribution if \(\alpha\approx 2\). The behavior of the distribution modes is also investigated numerically.
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 1452421
zbMATH Keywords: tail index
zbMATH Keywords: stable distribution
zbMATH Keywords: Hill estimator
zbMATH Keywords: mode
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1452421