### On Singular Wishart and Singular Multivariate Beta Distributions

Harald Uhlig
Source: Ann. Statist. Volume 22, Number 1 (1994), 395-405.

#### Abstract

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.

First Page:
Primary Subjects: 62H10
Secondary Subjects: 62E15
Full-text: Open access
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aos/1176325375
JSTOR: links.jstor.org
Digital Object Identifier: doi:10.1214/aos/1176325375
Mathematical Reviews number (MathSciNet): MR1272090
Zentralblatt MATH identifier: 0795.62052