Combining Independent Normal Mean Estimation Problems with Unknown Variances

Abstract

Let $X = (X_1, \cdots, X_p)^t$ be a $p$-variate normal random vector with unknown mean $\theta = (\theta_1, \cdots, \theta_p)^t$ and unknown positive definite diagonal covariance matrix $A$. Assume that estimates $V_i$ of the variances $A_i$ are available, and that $V_i/A_i$ is $\chi^2_{n_i}$. Assume also that all $X_i$ and $V_i$ are independent. It is desired to estimate $\theta$ under the quadratic loss $\lbrack\sum^p_{i=1} q_i(\delta_i - \theta_i)^2\rbrack/\lbrack\sum^p_{i=1} q_i A_i\rbrack,\quad\text{where} q_i > 0, i = 1, \cdots, p.$ Defining $W_i = V_i/(n_i - 2), W = (W_1, \cdots, W_p)^t$, and $\|X\|_{W^2} = \sum^p_{j=1} \lbrack X_{j^2}/(q_jW_j^2)\rbrack$, it is shown that under certain conditions on $r(X, W)$, the estimator given componentwise by $\delta_i(X, W) = (1 - r(X, W)/\lbrack\|X\|_{W^2}q_i W_i\rbrack)X_i$ is a minimax estimator of $\theta$. (The conditions on $r$ require $p \geqq 3$.) A good practical version of this estimator is also given.