The matrix angular central Gaussian distribution
Journal of Multivariate Analysis, 33 (2):
265--274 (1990)DOI: 10.1016/0047-259X(90)90050-R
Abstract
The Riemann space whose elements are $m k$ ($m k$) matrices $X$ such that $X′X = I_k$ is called the Stiefel manifold and denoted by $V_k,m$. Some distributions on $V_k,m$, e.g., the matrix Langevin (or von Mises-Fisher) and Bingham distributions and the uniform distribution, have been defined and discussed in the literature. In this paper, we present methods to construct new kinds of distributions on $V_k,m$ and discuss some properties of these distributions. We investigate distributions of the örientation" $H_Z = Z(Z′Z)^−1/2$ of an $m k$ random matrix $Z$. The general integral form of the density of $H_Z$ reduces to a simple mathematical form, when $Z$ has the matrix-variate central normal distribution with parameter $\Sigma$, an $m m $positive definite matrix. We may call this distribution the matrix angular central Gaussian distribution with parameter $\Sigma$, denoted by the MACG ($\Sigma$) distribution. The MACG distribution reduces to the angular central Gaussian distribution on the hypersphere for $k = 1$, which has been already known. Then, we are concerned with distributions of the orientation $H_Y$ of a linear transformation $Y = BZ$ of $Z$, where B is an m × m matrix such that $||B|| 0$. Utilizing properties of these distributions, we propose a general family of distributions of $Z$ such that $H_Z$ has the MACG ($\Sigma$) distribution.