A note about measures and Jacobians of singular random matrices

Abstract

This paper explains the differences between the densities and the Jacobians of the transforms of the same singular random matrices treated by several authors. Some comments on the results proposed by Srivastava [Singular Wishart and multivariate beta distributions, Ann. Statist. 31 (2003) 1537-1560] are presented. Definitions about a measure with respect to which a singular random matrix possesses a density are proposed. Finally two Jacobians of certain transforms under any of those measures are found.

The Density Functions of the Singular Quaternion Normal Matrix and the Singular Quaternion Wishart Matrix

“…Let X = X 1 + X 2 i + X 3 j + X 4 k ∈ m×n and X 1 X 2 X 3 X 4 are m × n matrices of functionally independent real variables. Define the volume of X as dX = dX 1 dX 2 dX 3 dX 4 , where dX s , s = 1 2 3 4, are defined in Mathai (1997) (or Díaz-García, 2007 3) when X s is functionally independent (or if X s is functionally dependent). The following lemma comes from Li and Xue (2009, Lemma 2.1).…”

“…It is expected that we can establish quaternion MIMO communication system and quaternion shape theory. Then we can apply the theory of the singular quaternion Wishart matrix to them like Ratnarajah et al (2003), Díaz-García andGutiérrez-Jáimez (2006), and.…”

Singular extended skew-elliptical distributions

“…Unfortunately, at present, some expressions can only be seen as theoretical results, because even in the univariate case there are important problems for classical estimation of parameters, see Azzalini (2005). In fact, some theoretical results for singular distributions of different kind of residuals (normalized, standardized, internally or externally studentized residual) cannot be established because the jacobians with respect to the Hausdorff measure are unknown; for example, no expressions for the simplest jacobian related with a linear transformation is given in literature, see Díaz-García (2007). In addition, from a Bayesian point of view we propose the a-priori parameter distributions of the corresponding matrix variate extended skew elliptically distribution and by using Definition 4.2 we avoid the parameter restrictions of Definition 4.1, see Press (1982, p. 253).…”

On the Geometry and Quantization of Manifolds of Positive Semi-Definite Matrices

“…Our calculation for the Pn manifold draws upon a different set of tools, which incidentally can be used to simplify the calculations in the Grassmann case of [3]. We use some recent developments in multivariate statistics [4], [5] to calculate the volume of the Pn manifold and then invoke a little-known result in Riemannian geometry by Gray [6] to compute the normalized volume of a ball.…”