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July, 1978 Deriving Unbiased Risk Estimators of Multinormal Mean and Regression Coefficient Estimators Using Zonal Polynomials
Jim Zidek
Ann. Statist. 6(4): 769-782 (July, 1978). DOI: 10.1214/aos/1176344251

Abstract

Unbiased risk estimators are derived for estimators in certain classes of equivariant estimators of multinormal matrix means, $\xi,$ and regression coefficients $\beta.$ In all cases the covariance matrix is unknown. The underlying method, a multivariate version of that of James and Stein (1960), uses zonal polynomial expansions for the distributions of noncentral statistics. This gives, in one case, the required generalization of the Pitman-Robbins representation of noncentral chi-square statistics including the appropriate multivariate Poisson law. In the other case, a multivariate negative binomial law emerges. The result for regression coefficients suggests a new minimax estimator and, essentially, an extension of Baranchik's result.

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Jim Zidek. "Deriving Unbiased Risk Estimators of Multinormal Mean and Regression Coefficient Estimators Using Zonal Polynomials." Ann. Statist. 6 (4) 769 - 782, July, 1978. https://doi.org/10.1214/aos/1176344251

Information

Published: July, 1978
First available in Project Euclid: 12 April 2007

zbMATH: 0379.62008
MathSciNet: MR478428
Digital Object Identifier: 10.1214/aos/1176344251

Subjects:
Primary: 62C15
Secondary: 62F10 , 62H10

Keywords: James-Stein estimator , minimax estimators , multivariate negative binomial , multivariate Poisson , multivariate regression , Pitman-Robbins representation , Unbiased risk estimators , zonal polynomials

Rights: Copyright © 1978 Institute of Mathematical Statistics

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Vol.6 • No. 4 • July, 1978
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