The Annals of Statistics

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

J. Berger, M. E. Bock, L. D. Brown, G. Casella, and L. Gleser

Full-text: Open access

Abstract

Let $X$ be an observation from a $p$-variate normal distribution $(p \geqq 3)$ with mean vector $\theta$ and unknown positive definite covariance matrix $\not\Sigma$. It is desired to estimate $\theta$ under the quadratic loss $L(\delta, \theta, \not\Sigma) = (\delta - \theta)^tQ(\delta - \theta)/\operatorname{tr} (Q\not\Sigma)$, where $Q$ is a known positive definite matrix. Estimators of the following form are considered: $\delta^c(X, W) = (I - c\alpha Q^{-1}W^{-1}/(X^tW^{-1}X))X,$ where $W$ is a $p \times p$ random matrix with a Wishart $(\not\Sigma, n)$ distribution (independent of $X$), $\alpha$ is the minimum characteristic root of $(QW)/(n - p - 1)$ and $c$ is a positive constant. For appropriate values of $c, \delta^c$ is shown to be minimax and better than the usual estimator $\delta^0(X) = X$.

Article information

Source
Ann. Statist., Volume 5, Number 4 (1977), 763-771.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176343898

Digital Object Identifier
doi:10.1214/aos/1176343898

Mathematical Reviews number (MathSciNet)
MR443156

Zentralblatt MATH identifier
0356.62009

JSTOR
links.jstor.org

Subjects
Primary: 62C99: None of the above, but in this section
Secondary: 62F10: Point estimation 62H99: None of the above, but in this section

Keywords
Minimax normal mean quadratic loss unknown covariance matrix Wishart risk function

Citation

Berger, J.; Bock, M. E.; Brown, L. D.; Casella, G.; Gleser, L. Minimax Estimation of a Normal Mean Vector for Arbitrary Quadratic Loss and Unknown Covariance Matrix. Ann. Statist. 5 (1977), no. 4, 763--771. doi:10.1214/aos/1176343898. https://projecteuclid.org/euclid.aos/1176343898


Export citation