  In this paper, we consider the problem of determining the presence of a given
signal in a high-dimensional observation with unknown covariance matrix by
using an adaptive matched filter. Traditionally such filters are formed from
the sample covariance matrix of some given training data, but, as is
well-known, the performance of such filters 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 asymptotically
detection-theoretic optimal within a general shrinkage class inspired by C.
Stein, and we give consistent estimates for conditional false-alarm and
detection rate of the corresponding adaptive matched filter.
