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{1176325375,
author = {Uhlig, Harald},
title = {On Singular Wishart and Singular Multivariate Beta Distributions},
journal = {Ann. Statist.},
volume = {22},
number = {1},
year = {1994},
pages = { 395-405},
language = {en},
url = {http://dml.mathdoc.fr/item/1176325375}
}
Uhlig, Harald. On Singular Wishart and Singular Multivariate Beta Distributions. Ann. Statist., Tome 22 (1994) no. 1, pp. 395-405. http://gdmltest.u-ga.fr/item/1176325375/