The Annals of Statistics

The Non-Singularity of Generalized Sample Covariance Matrices

Morris L. Eaton and Michael D. Perlman

Full-text: Open access

Abstract

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 $F \subset R^p$.

Article information

Source
Ann. Statist., Volume 1, Number 4 (1973), 710-717.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176342465

Digital Object Identifier
doi:10.1214/aos/1176342465

Mathematical Reviews number (MathSciNet)
MR341745

Zentralblatt MATH identifier
0261.62037

JSTOR
links.jstor.org

Subjects
Primary: 62H10: Distribution of statistics
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 15A03: Vector spaces, linear dependence, rank

Keywords
Independent random vectors nonsingularity of random matrices sample covariance matrix linear manifolds flats

Citation

Eaton, Morris L.; Perlman, Michael D. The Non-Singularity of Generalized Sample Covariance Matrices. Ann. Statist. 1 (1973), no. 4, 710--717. doi:10.1214/aos/1176342465. https://projecteuclid.org/euclid.aos/1176342465


Export citation