The Annals of Statistics

Minimax Estimators of a Normal Mean Vector for Arbitrary Quadratic Loss and Unknown Covariance Matrix

Leon Jay Gleser

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Abstract

The problem of finding classes of estimators which dominate the usual estimator $X$ of the mean vector $\mu$ of a $p$-variate normal distribution $(p \geq 3)$ under general quadratic loss is analytically difficult in cases where the covariance matrix is unknown. Estimators of $\mu$ in this case depend upon $X$ and an independent Wishart matrix $W$. In the present paper, integration-by-parts methods for both the multivariate normal and Wishart distributions are combined to yield unbiased estimates of risk difference (versus $X$) for certain classes of estimators, defined indirectly through a "seed" function $h(X, W)$. An application of this technique produces a new class of minimax estimators of $\mu$.

Article information

Source
Ann. Statist., Volume 14, Number 4 (1986), 1625-1633.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176350184

Digital Object Identifier
doi:10.1214/aos/1176350184

Mathematical Reviews number (MathSciNet)
MR868326

Zentralblatt MATH identifier
0613.62004

JSTOR
links.jstor.org

Subjects
Primary: 62C20: Minimax procedures
Secondary: 62F11 62H99: None of the above, but in this section 62J07: Ridge regression; shrinkage estimators

Keywords
Integration-by-parts identities unbiased estimates of risk difference Wishart distribution

Citation

Gleser, Leon Jay. Minimax Estimators of a Normal Mean Vector for Arbitrary Quadratic Loss and Unknown Covariance Matrix. Ann. Statist. 14 (1986), no. 4, 1625--1633. doi:10.1214/aos/1176350184. https://projecteuclid.org/euclid.aos/1176350184


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