Uhlig, Harald On singular Wishart and singular multivariate beta distributions. (English) Zbl 0795.62052 Ann. Stat. 22, No. 1, 395-405 (1994). Summary: This paper extends the study of Wishart and multivariate beta distributions to the singular case, where the rank is below the dimensionality. The usual conjugacy is extended to this case. A volume element on the space of positive semidefinite \(m\times m\) matrices of rank \(n<m\) is introduced and some transformation properties established. The density function is found for all rank-\(n\) Wishart distributions as well as the rank-1 multivariate beta distribution. To do that, the Jacobian for the transformation to the singular value decomposition of general \(m\times n\) matrices is calculated. The results in this paper are useful in particular for updating a Bayesian posterior when tracking a time-varying variance-covariance matrix. Cited in 4 ReviewsCited in 72 Documents MSC: 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62H10 Multivariate distribution of statistics Keywords:updating Bayesian posteriors; Stiefel manifold; multivariate beta distributions; singular case; conjugacy; volume element; transformation properties; Wishart distributions; singular value decomposition; time- varying variance-covariance matrix PDFBibTeX XMLCite \textit{H. Uhlig}, Ann. Stat. 22, No. 1, 395--405 (1994; Zbl 0795.62052) Full Text: DOI