The Annals of Statistics

Empirical Bayes Estimation of the Multivariate Normal Covariance Matrix

L. R. Haff

Full-text: Open access

Abstract

Let $\mathbf{S}_{p \times p}$ have a Wishart distribution with scale matrix $\Sigma$ and $k$ degrees of freedom. Estimators of $\Sigma$ are given for each of the loss functions $L_1(\hat{\Sigma}, \Sigma) = \operatorname{tr} (\hat{\Sigma}\Sigma^{-1}) - \log \det (\hat{\Sigma}\Sigma^{-1}) - p$ and $L_2(\hat{\Sigma}, \Sigma) = \operatorname{tr} (\hat{\Sigma}\Sigma^{-1} - I)^2$. The obvious estimators of $\Sigma$ are the scalar multiples of $\mathbf{S}$, i.e., $a\mathbf{S}$ where $0 < a \leqslant 1/k$. (Recall that $(1/k)\mathbf{S}$ is unbiased.) For each problem $(\Sigma, \hat{\Sigma}, L_i), i = 1, 2$, we provide empirical Bayes estimators which dominate $a\mathbf{S}$ by a substantial amount. It is seen that the uniform reduction in the risk function determined by $L_2$ is at least $100(p + 1)/(k + p + 1){\tt\%}$. Dominance results for $L_1$ and $L_2$ were first given by James and Stein.

Article information

Source
Ann. Statist., Volume 8, Number 3 (1980), 586-597.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345010

Digital Object Identifier
doi:10.1214/aos/1176345010

Mathematical Reviews number (MathSciNet)
MR568722

Zentralblatt MATH identifier
0441.62045

JSTOR
links.jstor.org

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

Keywords
Covariance matrix empirical Bayes estimators unbiased estimation of risk function

Citation

Haff, L. R. Empirical Bayes Estimation of the Multivariate Normal Covariance Matrix. Ann. Statist. 8 (1980), no. 3, 586--597. doi:10.1214/aos/1176345010. https://projecteuclid.org/euclid.aos/1176345010


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