Maximum likelihood estimates and likelihood ratio criteria for location and scale parameters of the multivariate \(l_ 1\)-norm symmetric distributions
type: Item
instance of: scholarly article
title: Maximum likelihood estimates and likelihood ratio criteria for location and scale parameters of the multivariate \(l_ 1\)-norm symmetric distributions
zbMATH Open document ID: 0687.62041
DOI: 10.1007/BF02018709
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 1988-01-01T00:00:00Z
review text: The family of matrix \(\ell_ 1\)-norm symmetric distributions is defined as the family of distributions of \(\underset \tilde{} Z\) of \(n\times p\), where \(\underset \tilde{} Z=\underset \tilde{} U \underset \tilde{} R\), \(\underset \tilde{} U\) is a matrix whose columns are uniformly distributed over a set \(B_ n=\{\underset \tilde{} b:\) \(\underset \tilde{} b\in R^ n_+\), \(\| b\| =1\}\), \(\underset \tilde{} R\) is diagonal, \(\underset \tilde{} R=diag(r_ 1,...,r_ p)\geq 0\), and \(\underset \tilde{} U\) and \(\underset \tilde{} R\) are independent. Parameters are introduced by setting \(\underset \tilde{} W=\underset \tilde{} M+\underset \tilde{} Z\underset \tilde{} A\), where \(\underset \tilde{} A=diag(a_ 1,...,a_ p)\geq \underset \tilde{} O\). Basic properties of this and related families are given by the authors in previous work, e.g. J. Multivariate Anal. 24, No.1, 109-122 (1988; Zbl 0635.62035)]. The authors show that if g is the probabiity density function of \(\| \underset \tilde{} Z\|\), the maximum likelihood estimator of \(\underset \tilde{} M\) is independent of g, the MLE of \(\underset \tilde{} A\) is invariant except for a constant multiplier (depending on g), and for some hypotheses the corresponding likelihood ratio criteria and their null distributions are independent of g. The authors emphasize that (under certain conditions) these results have the same form as those for independent exponential variables.
Mathematics Subject Classification ID: 62H12
Mathematics Subject Classification ID: 62H10
zbMATH DE Number: 4126499
zbMATH Keywords: \(l_ 1\)-norm symmetric distributions
zbMATH Keywords: maximum likelihood estimator
zbMATH Keywords: likelihood ratio criteria
zbMATH Keywords: null distributions
zbMATH Keywords: independent exponential variables
author: Bi Qi Fang
author: Kai-Tai Fang
MaRDI profile type: Publication
cites work: Some families of multivariate symmetric distributions related to exponential distribution
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cites work: Maximum-likelihood estimates and likelihood-ratio criteria for multivariate elliptically contoured distributions
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description: scientific article; zbMATH DE number 4126499