On statistical tests for a simple hypotheses in the case of multy-dimensional normal distribution

The optimal statistical test for the choice of one of two simple hypothesis in the case of multi-dimensional normal distribution is very complicated in practical applications. One can construct a simplified test neglecting the dependence of the components of the normal variate. In this paper the difference between the probabilities of the second kind errors of the optimal and simplified tests is investigated. A necessary and sufficient condition for this difference to be equal to zero as well as some estimates, of this difference are given. The case when the parameters of the normal distribution are a priori unknown is considered separately.