The Annals of Statistics

On Singular Wishart and Singular Multivariate Beta Distributions

Harald Uhlig

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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: 11 April 2007

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

Subjects
Primary: 62H10: Distribution of statistics
Secondary: 62E15: Exact distribution theory

Keywords
Wishart distribution beta distribution Stiefel manifold singular matrix distributions conjugacy

Citation

Uhlig, Harald. On Singular Wishart and Singular Multivariate Beta Distributions. The Annals of Statistics 22 (1994), no. 1, 395--405. doi:10.1214/aos/1176325375. http://projecteuclid.org/euclid.aos/1176325375.


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