We consider mixtures of finitely supported discrete distributions, that is, distributions that put their total mass on a finite set. We assume that this set is of the form \(\{0, 1, \ldots , K\}\) for some known integer K. We show that if the true mixture lies in the interior of the class of mixtures under study, then the non-parametric maximum likelihood estimator equals the empirical estimator with probability one when the sample size n is large enough. Additionally, we provide an explicit lower bound on n so that this equality holds with high probability. For mixtures of Binomials, we use the already known characterization of the interior through the index number of the true Binomial mixture to construct a simple testing procedure based on determinants of some Hankel matrices. This test enables to find out in a straightforward way whether the number of components of the true mixture is above a certain threshold, and hence whether the maximum likelihood estimator and the empirical estimator are equal. The usefulness of the testing procedure is demonstrated through a simulation study where different values of K and number of components are considered.
