Asymptotic expansions for large deviation probabilities of noncentral generalized chi-square distributions
type: Item
instance of: scholarly article
title: Asymptotic expansions for large deviation probabilities of noncentral generalized chi-square distributions
zbMATH Open document ID: 0976.60037
author: J. Schumacher
author: Wolf-Dieter Richter
published in: Journal of Multivariate Analysis
publication date: 2001-12-10T00:00:00Z
review text: Several asymptotic representations for large deviation probabilities are known in the literaure for the one-dimensional as well as the multi-dimensional case [cf. \textit{H. E. Daniels}, Int. Stat. Rev. 55, 37-48 (1987; Zbl 0614.62016), \textit{C. Ittrich, D. Krause} and \textit{W.-D. Richter}, Statistics 34, No. 1, 53-101 (2000), \textit{K. Breitung} and \textit{W.-D. Richter}, J. Multivariate Anal. 58, No. 1, 1-20 (1996; Zbl 0864.41026)]. The authors solve the problem of evaluating the tail probabilities \(1-\)CQ\((k,\delta^2;g)\) where CQ\((k,\delta^2;g)\) denotes the noncentral \(g\)-generalized chi-square distribution with \(k\) d.f. and non-centrality parameter \(\delta^2\) for shperically distributed random vectors. Using a geometric representation formula for the measure, \(1-\)CQ\((k,\delta^2;g)\) gives a Laplace type integral representation which can be expanded in a series. A modified version of Watson's lemma finally leads to the desired asymptotic expansion for the tail probabilities \(1-\)CQ\((k,\delta^2;g)\). The authors apply their result for Kotz type densities and present some numerical calculations.
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 62E17
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62H30
zbMATH DE Number: 1563426
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: noncentral distribution
zbMATH Keywords: large deviations
zbMATH Keywords: Watson's lemma
zbMATH Keywords: chi-square distribution
zbMATH Keywords: geometric representation
zbMATH Keywords: oscillators
reviewed by: Wolfgang zu Castell
MaRDI profile type: Publication
full work available at URL: jmva.2000.1899
OpenAlex ID: W2079587349
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DOI: 10.1006/JMVA.2000.1899
description: scientific article; zbMATH DE number 1563426