On detection of Gaussian stochactic vectors

The problem of minimax detection of a Gaussian random signal vector in white Gaussian additive noise has been considered. We suppose that an unknown vector $\boldsymbol{\sigma}$ of the signal vector intensities belongs to the given set ${\mathcal E}$. We have investigated when it is possible to replace the set ${\mathcal E}$ (and, in particular, by a single point $\boldsymbol{\sigma}_{0}$) by a smaller set ${\mathcal E}_{0}$ without quality loss.