An asymptotically optimal test for the number of components of a mixture of probability distributions

The problem of statistical testing of hypotheses concerning the number of components in a mixture of probability distributions is considered. An asymptotically most powerful test is presented. Under rather weak conditions, the limit distributions, power loss, and the asymptotic deficiency are found. The application of this test to verification of hypotheses concerning the number of components in a mixture of uniform, normal, and gamma distributions is considered in detail.