## The Annals of Statistics

### On Singular Wishart and Singular Multivariate Beta Distributions

Harald Uhlig

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 11 April 2007

http://projecteuclid.org/euclid.aos/1176325375

Digital Object Identifier
doi:10.1214/aos/1176325375

Mathematical Reviews number (MathSciNet)
MR1272090

Zentralblatt MATH identifier
0795.62052

JSTOR