The Annals of Statistics

Minimax Estimation of a Normal Mean Vector When the Covariance Matrix is Unknown

Leon Jay Gleser

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Let $X$ be an observation from a $p$-variate normal distribution $(p \geqslant 3)$ with mean vector $\theta$ and unknown positive definite covariance matrix $\sum$. We wish to estimate $\theta$ under the quadratic loss $L(\delta; \theta, \sum) = \lbrack\mathrm{tr}(Q\sum)\rbrack^{-1}(\delta - \theta)'Q(\delta - \theta)$, where $Q$ is a known positive definite matrix. Estimators of the following form are considered: $\delta_{k, h}(X, W) = \lbrack I - kh(X'W^{-1}X)\lambda_1(QW/n^\ast)Q^{-1}W^{-1}\rbrack X,$ where $W : p \times p$ is observed independently of $X$ and has a Wishart distribution with $n$ degrees of freedom and parameter $\sum, \lambda_1(A)$ denotes the minimum characteristic root of $A$, and $h(t): \lbrack 0, \infty) \rightarrow \lbrack 0, \infty)$ is absolutely continuous with respect to Lebesgue measure, is nonincreasing, and satisfies the additional requirements that $th(t)$ is nondecreasing and $\sup_{t \geqslant 0}th(t) = 1$. With $h(t) = t^{-1}$, the class $\delta_{k, h}$ specializes to that considered by Berger, Bock, Brown, Casella and Gleser (1977). For the more general class considered in the present paper, it is shown that there is an interval $\lbrack 0, k_{n, p}\rbrack$ of values of $k$ (which may be degenerate for small values of $n - p)$ for which $\delta_{k, h}$ is minimax and dominates the usual estimator $\delta_0 \equiv X$ in risk.

Article information

Ann. Statist., Volume 7, Number 4 (1979), 838-846.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


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

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


Gleser, Leon Jay. Minimax Estimation of a Normal Mean Vector When the Covariance Matrix is Unknown. Ann. Statist. 7 (1979), no. 4, 838--846. doi:10.1214/aos/1176344733.

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