  We consider the problem of detecting (testing) Gaussian stochastic sequences
(signals) with imprecisely known means and covariance matrices. The alternative
is independent identically distributed zero-mean Gaussian random variables with
unit variances. For a given false alarm (1st-kind error) probability, the
quality of minimax detection is given by the best miss probability (2nd-kind
error probability) exponent over a growing observation horizon. We explore the
maximal set of means and covariance matrices (composite hypothesis) such that
its minimax testing can be replaced with testing a single particular pair
consisting of a mean and a covariance matrix (simple hypothesis) without
degrading the detection exponent. We completely describe this maximal set.
