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March, 1994 On Singular Wishart and Singular Multivariate Beta Distributions
Harald Uhlig
Ann. Statist. 22(1): 395-405 (March, 1994). DOI: 10.1214/aos/1176325375


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.


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Harald Uhlig. "On Singular Wishart and Singular Multivariate Beta Distributions." Ann. Statist. 22 (1) 395 - 405, March, 1994.


Published: March, 1994
First available in Project Euclid: 11 April 2007

zbMATH: 0795.62052
MathSciNet: MR1272090
Digital Object Identifier: 10.1214/aos/1176325375

Primary: 62H10
Secondary: 62E15

Keywords: Beta distribution , conjugacy , singular matrix distributions , Stiefel manifold , Wishart distribution

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.22 • No. 1 • March, 1994
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