The Annals of Statistics

Singular Wishart and multivariate beta distributions

M.S. Srivastava

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Abstract

In this article, we consider the case when the number of observations n is less than the dimension p of the random vectors which are assumed to be independent and identically distributed as normal with nonsingular covariance matrix. The central and noncentral distributions of the singular Wishart matrix $S=XX'$, where X is the $p \times n$ matrix of observations are derived with respect to Lebesgue measure. Properties of this distribution are given. When the covariance matrix is singular, pseudo singular Wishart distribution is also derived. The result is extended to any distribution of the type $f(XX')$ for the central case. Singular multivariate beta distributions with respect to Lebesgue measure are also given.

Article information

Source
Ann. Statist., Volume 31, Number 5 (2003), 1537-1560.

Dates
First available in Project Euclid: 9 October 2003

Permanent link to this document
https://projecteuclid.org/euclid.aos/1065705118

Digital Object Identifier
doi:10.1214/aos/1065705118

Mathematical Reviews number (MathSciNet)
MR2012825

Zentralblatt MATH identifier
1042.62051

Subjects
Primary: 62H10: Distribution of statistics
Secondary: 62E15: Exact distribution theory

Keywords
Jacobian of transformations normal distribution pseudo Wishart singular noncentral Wishart Stiefel manifold

Citation

Srivastava, M.S. Singular Wishart and multivariate beta distributions. Ann. Statist. 31 (2003), no. 5, 1537--1560. doi:10.1214/aos/1065705118. https://projecteuclid.org/euclid.aos/1065705118


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