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Pseudo-inverse multivariate/matrix-variate distributions

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  • Zhang, Zhihua

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, pseudo-inverse matrix-variate normal distributions and pseudo-inverse Wishart distributions.

Suggested Citation

  • Zhang, Zhihua, 2007. "Pseudo-inverse multivariate/matrix-variate distributions," Journal of Multivariate Analysis, Elsevier, vol. 98(8), pages 1684-1692, September.
    • Handle: RePEc:eee:jmvana:v:98:y:2007:i:8:p:1684-1692
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    References listed on IDEAS

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    1. José Díaz-García & Ramón Gutiérrez-Jáimez, 2005. "Functions of singular random matrices with applications," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 14(2), pages 475-487, December.
    2. Díaz-García, José A. & Jáimez, Ramón Gutierrez & Mardia, Kanti V., 1997. "Wishart and Pseudo-Wishart Distributions and Some Applications to Shape Theory," Journal of Multivariate Analysis, Elsevier, vol. 63(1), pages 73-87, October.
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