Estimating Covariance Matrices

Abstract

Let $S_1$ and $S_2$ be two independent $p \times p$ Wishart matrices with $S_1 \sim W_p(\sum_1, n_1)$ and $S_2 \sim W_p(\sum_2, n_2)$. We wish to estimate $(\sum_1, \sum_2)$ under the loss function $L(\hat{\sum}_1, \hat{\sum}_2; \sum_1, \sum_2) = \sum_i\{\operatorname{tr}(\sum^{-1}_i \hat{\sum}_i) - \log|\sum^{-1}_i\hat{\sum}_i| - p\}$. Our approach is to first utilize the principle of invariance to narrow the class of estimators under consideration to the equivariant ones. The unbiased estimates of risk of these estimators are then computed and promising estimators are derived from them. A Monte Carlo study is also conducted to evaluate the risk performances of these estimators. The results of this paper extend those of Stein, Haff, Dey and Srinivasan from the one sample problem to the two sample one.