Recent asymptotic results in testing for mixtures

Inferences concerning the number of components in a mixture distribution are often required. These can be performed under the framework of the generalized likelihood ratio test. The classical result giving a chi-squared asymptotic distribution in general does not apply, indeed the limiting distribution of the corresponding test statistic has long remained a mystery. The characterization of the asymptotic distribution in a general setting has been previously derived under a separation condition. The relaxation of the separation condition, the calculation of the percentile points and asymptotic power, both in the case of a bounded and an unbounded parameter set can be obtained. An overview of the very recent asymptotic results in the problem of testing homogeneity against a two-component mixture is provided. Illustrations of new and known results are presented.