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June, 1985 Best Invariant Estimation of a Direction Parameter
T. W. Anderson, Charles Stein, Asad Zaman
Ann. Statist. 13(2): 526-533 (June, 1985). DOI: 10.1214/aos/1176349536

Abstract

Let $X$ be an $n \times k$ random matrix whose coordinates are independently normally distributed with common variance $\sigma^2$ and means given by $EX = e\mu' + \theta\lambda',$ where $e$ is the vector in $R^n$ having all coordinates equal to $1, \theta \in R^n,$ and $\mu, \lambda \in R^k$ with $\sum^k_{j = 1} \lambda^2_j = 1.$ The problem is to estimate $\lambda$, say by $\hat{\lambda},$ with loss function $1 - (\lambda'\hat{\lambda})^2$ when $\mu, \theta,$ and $\sigma^2$ are unknown. It is shown that the largest principal component of $X'X - (1/n)X'ee'X$ is the best estimator invariant under rotations in $R^k$ and rotations in $R^n$ leaving $e$ invariant and is admissible.

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T. W. Anderson. Charles Stein. Asad Zaman. "Best Invariant Estimation of a Direction Parameter." Ann. Statist. 13 (2) 526 - 533, June, 1985. https://doi.org/10.1214/aos/1176349536

Information

Published: June, 1985
First available in Project Euclid: 12 April 2007

zbMATH: 0583.62044
MathSciNet: MR790554
Digital Object Identifier: 10.1214/aos/1176349536

Subjects:
Primary: 62C15
Secondary: 62F10

Rights: Copyright © 1985 Institute of Mathematical Statistics

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Vol.13 • No. 2 • June, 1985
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