Maximum likelihood estimates and likelihood ratio criteria for location and scale parameters of the multivariate \(l_ 1\)-norm symmetric distributions
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.
scientific article; zbMATH DE number 4126499