Pseudo-inverse multivariate/matrix-variate distributions

Abstract

The Moore-Penrose inverse of a singular or nonsquare matrix is not only existent but also unique. In this paper, we derive the Jacobian of the transformation from such a matrix to the transpose of its Moore-Penrose inverse. Using this Jacobian, we investigate the distribution of the Moore-Penrose inverse of a random matrix and propose the notion of pseudo-inverse multivariate/matrix-variate distributions. For arbitrary multivariate or matrix-variate distributions, we can develop the corresponding pseudo-inverse distributions. In particular, we present pseudo-inverse multivariate normal distributions, pseudo-inverse Dirichlet distributions, pseudoinverse matrix-variate normal distributions and pseudo-inverse Wishart distributions.

A note about measures and Jacobians of singular random matrices

“…A similar situation appear in the following cases: i) When we consider multiplicity of the singular values in the SVD; such set of matrices will be denoted by X ∈ L l m,N (q), q ≥ l; or by X ∈ L + m,N (q, l) q ≥ l; ii) If we consider multiplicity in the eigenvalues; the corresponding set of matrices will be denoted by A ∈ S + m (q, l), q ≥ l; iii) And if A is nonpositive definite. Thus, unfortunately, we must qualify as incorrect the asseverations of Zhang (2007) about the validity of his Lemmas 2, 3 and consequences, under multiplicity assumptions of singular values and eigenvalues.…”

On the mean and dispersion of the Moore-Penrose generalized inverse of a Wishart matrix

“…Concerning the density of a Moore-Penrose inverse there are some results available when a matrix Z, has full column rank. In this case, Z + = (Z Z) −1 Z has a density |Z + (Z + ) | −p f (Z), where f (Z) is the density for Z (see [2,5,10,11]). From this expression, in principle, the density for W + can be found and advanced direct calculations of Jacobians leads to the density for W + (see [5,11]).…”

Singular matrix variate Birnbaum-Saunders distribution under elliptical models

“…• As a consequence, when alternative approaches of literature are used in the setting of the present work, it should be noted that inconsistent algebraic, probabilistic and conceptual results are obtained. Furthermore, in those papers the approach followed in this work is validated by different authors, see Zhang [28], Díaz-García [8], and Bondar and Okrin [4], among others.…”

“…Now, when the research moves to the singular case, the problems are greater, because the distributions do not exist with respect the Lebesgue measure, see Khatri [22]. For a unified approach, summarising a number of singular distributions in a wider spectra, see Khatri [22], Uhlig [26], Rao [25], Díaz-García et al [16], Díaz-García and González-Farías [13,14], Díaz-García and Gutiérrez [17,18,19], Zhang [28], Bondar and Okrin [4] and the references therein.…”