Moments of generalized Wishart distributions
type: Item
instance of: scholarly article
title: Moments of generalized Wishart distributions
zbMATH Open document ID: 0877.62051
author: Chi Song Wong
author: D. S. Liu
published in: Journal of Multivariate Analysis
publication date: 1997-12-14T00:00:00Z
review text: Moments of a distribution are of interest partially because they can be used to represent certain statistical models, to carry out statistical inference, and to approximate the given distribution. Moment calculation has, however, been influenced by results on matrix algebra and derivatives. In fact, many results on matrix algebra and derivatives have been motivated by the problem of finding moments and for other statistical results. In Sections 2 and 3, we relate moments of a random matrix \(Y\) to tensor differential forms and permutations without assuming that \(Y\) is normal or left elliptically contoured. Examples are given for two large classes of distributions. The first class consists of the generalized Wishart distributions induced by an \(s\times t\) random matrix \(Y\) with covariance \(I_s \otimes \Sigma\). The second class consists of the generalized complex Wishart distributions induced by a \((2n) \times p\) random matrix \(Y\) with covariance \(I_n\otimes \left(\begin{smallmatrix} \Sigma_1 & - \Sigma_2 \\ \Sigma_2 & \Sigma_3\end{smallmatrix} \right)\). \textit{N. R. Goodman} [see Ann. Math. Stat. 34, 152-177 (1963; Zbl 0122.36903)] and many other authors considered the case where \(Y\) is normal, the covariance is nonsingular, and \(\Sigma_1 =\Sigma_3\). Our generalized complex Wishart distributions combine real Wishart distributions and the complex Wishart distributions of Goodman into a much larger class. In Section 4, we find the \(q\)th order moments of noncentral generalized Wishart distributions induced by left elliptically contoured distributions. The formula contains \(\sum^q_{v=0} (2q)!/[v! 2^v(2q-2v)!]\) terms and a certain degree of complication is, therefore, inevitable. We shall derive this formula in a simple way and represent it in a transparent form.
Mathematics Subject Classification ID: 62H05
Mathematics Subject Classification ID: 62H10
Mathematics Subject Classification ID: 62H99
Mathematics Subject Classification ID: 53A45
zbMATH DE Number: 739446
zbMATH Keywords: characteristic function
zbMATH Keywords: generalized real and complex Wishart distributions
zbMATH Keywords: higher order moments
zbMATH Keywords: inner product
zbMATH Keywords: Kronecker (tensor) product
zbMATH Keywords: multivariate left elliptically contoured distribution
zbMATH Keywords: permutation and reindexing
zbMATH Keywords: tensor differential form
zbMATH Keywords: zonal polynomial
MaRDI profile type: Publication
full work available at URL: jmva.1995.1014
OpenAlex ID: W2041309907
DOI: 10.1006/JMVA.1995.1014
description: scientific article; zbMATH DE number 739446