The Jacobians of matrix transformation about singular random matrices and its applications

Fei Li

Communications in Statistics - Theory and Methods, 2021, vol. 50, issue 20, 4718-4732

Abstract: In this paper, we give the Jacobians of matrix transformations Yq×1=Aq×pXp×1,Ym×n=Am×pXp×qBq×n and W˜q×q=Aq×pWp×pA′q×p, when the constant matrices A and B are singular rectangular matrices, not limited to square matrices, and the random variables in the random matrices X, Y can be linearly dependent, W and W˜ are singular random matrices. Furthermore, some important properties about singular multivariate Normal distribution and singular Wishart distribution are proved by using these Jacobians. Finally, the density function of AW−A′ and the distribution of generalized Hotelling’s T2 statistic are obtained.

Date: 2021
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DOI: 10.1080/03610926.2020.1723634

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