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December, 1989 Bounds for the Distribution of the Generalized Variance
Louis Gordon
Ann. Statist. 17(4): 1684-1692 (December, 1989). DOI: 10.1214/aos/1176347387

Abstract

Let $D_{p,m}$ be the determinant of the sample covariance matrix for $m + p + 1$ observations from a $p$-variate normal population having identity covariance matrix. We give bounds for the distribution of $D_{p,m}$ in terms of various chi-squared distribution functions. Let $F(\cdot \mid \nu)$ denote the chi-squared distribution function on $\nu$ degrees of freedom. We bound $P\{p(D_{p,m})^{1/p} > t\}$ above by $1 - F(t \mid p(m + 1) + \frac{1}{2}(p - 1)(p - 2))$ and below by $1 - F(t \mid p(m + 1))$. We give two more bounds involving chi-squared distributions. The proofs use a stochastic analog to the Gauss multiplication theorem.

Citation

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Louis Gordon. "Bounds for the Distribution of the Generalized Variance." Ann. Statist. 17 (4) 1684 - 1692, December, 1989. https://doi.org/10.1214/aos/1176347387

Information

Published: December, 1989
First available in Project Euclid: 12 April 2007

zbMATH: 0695.62128
MathSciNet: MR1026305
Digital Object Identifier: 10.1214/aos/1176347387

Subjects:
Primary: 62H10
Secondary: 33A15

Rights: Copyright © 1989 Institute of Mathematical Statistics

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Vol.17 • No. 4 • December, 1989
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