  In this paper, we consider the general problem of testing the mean of two
high-dimensional distributions with a common, unknown covariance using a linear
classifier. Traditionally such a classifier is formed from the sample
covariance matrix of some given training data, but, as is well-known, the
performance of this classifier is poor when the number of training data $n$ is
not much larger than the data dimension $p$. We thus seek a covariance
estimator to replace sample covariance. To account for the fact that $n$ and
$p$ may be of comparable size, we adopt the "large-dimensional asymptotic
model" in which $n$ and $p$ go to infinity in a fixed ratio. Under this
assumption, we identify a covariance estimator that is detection-theoretic
optimal within the general shrinkage class of C. Stein, and we give consistent
estimates for the corresponding classifier's type-I and type-II errors.
