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July, 1973 The Non-Singularity of Generalized Sample Covariance Matrices
Morris L. Eaton, Michael D. Perlman
Ann. Statist. 1(4): 710-717 (July, 1973). DOI: 10.1214/aos/1176342465


Let $X = (X_1, \cdots, X_n)$ where the $X_i: p \times 1$ are independent random vectors, and let $A: n \times n$ be positive semi-definite symmetric. This paper establishes necessary and sufficient conditions that the random matrix $XAX'$ be positive definite w.p.1. The results are applied to cases where $A$ has a particular form or $X_1, \cdots, X_n$ are i.i.d. In particular, it is shown that in the i.i.d. case, the sample covariance matrix $\sigma(X_i - \bar{X})(X_i - \bar{X})'$ is positive definite w.p. 1 $\operatorname{iff} P\lbrack X_1 \in F\rbrack = 0$ for every proper flat $F \subset R^p$.


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Morris L. Eaton. Michael D. Perlman. "The Non-Singularity of Generalized Sample Covariance Matrices." Ann. Statist. 1 (4) 710 - 717, July, 1973.


Published: July, 1973
First available in Project Euclid: 12 April 2007

zbMATH: 0261.62037
MathSciNet: MR341745
Digital Object Identifier: 10.1214/aos/1176342465

Primary: 62H10
Secondary: 15A03 , 60D05

Keywords: flats , independent random vectors , linear manifolds , nonsingularity of random matrices , Sample covariance matrix

Rights: Copyright © 1973 Institute of Mathematical Statistics


Vol.1 • No. 4 • July, 1973
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