The Annals of Statistics

On Singular Wishart and Singular Multivariate Beta Distributions

Harald Uhlig

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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

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

First available in Project Euclid: 11 April 2007

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Zentralblatt MATH identifier


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

Wishart distribution beta distribution Stiefel manifold singular matrix distributions conjugacy


Uhlig, Harald. On Singular Wishart and Singular Multivariate Beta Distributions. Ann. Statist. 22 (1994), no. 1, 395--405. doi:10.1214/aos/1176325375.

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