Two-sided Estimates for the Sum of Probabilities of Errors in the Multiple Hypothesis Testing Problem with Finite Number of Hypotheses on a Nonhomogeneous Sample

We obtain two-sided estimates for the weighted sum of probabilities of errors in the multiple hypothesis testing problem with finite number of hypotheses on a nonhomogeneous sample of size $n$. The obtained upper and lower estimates are shown to converge to zero exponentially fast with increasing $n$ in a wide class of cases. The results obtained can be used for deriving two-sided estimates for the size of a sample required for multiple hypothesis testing.