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May, 1976 An Improved Estimator of the Generalized Variance
R. W. Shorrock, J. V. Zidek
Ann. Statist. 4(3): 629-638 (May, 1976). DOI: 10.1214/aos/1176343470


A multivariate extension is made of Stein's result (1964) on the estimation of the normal variance. Here the generalized variance $|\Sigma|$ is being estimated from a Wishart random matrix $S: p \times p \sim W(n, \Sigma)$ and an independent normal random matrix $X: p \times k \sim N(\xi, \Sigma \otimes 1_k)$ with $\xi$ unknown. The main result is that the minimax, best affine equivariant estimator $((n + 2 - p)!/(n + 2)!)|S|$ is dominated by $\min\{((n + 2 - p)!/(n + 2)!)|S|, ((n + k + 2 - p)!/(n + k + 2)!)|S + XX'|\}$. It is obtained by a variant of Stein's method which exploits zonal polynomials.


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R. W. Shorrock. J. V. Zidek. "An Improved Estimator of the Generalized Variance." Ann. Statist. 4 (3) 629 - 638, May, 1976.


Published: May, 1976
First available in Project Euclid: 12 April 2007

zbMATH: 0353.62039
MathSciNet: MR411034
Digital Object Identifier: 10.1214/aos/1176343470

Primary: 62F10
Secondary: 62H99

Keywords: equivariant , multivariate normal matrix , Noncentral Wishart matrix , zonal polynomials

Rights: Copyright © 1976 Institute of Mathematical Statistics

Vol.4 • No. 3 • May, 1976
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