Testing the Order of a Finite Mixture

The order is an important parameter in applications of finite mixture models. Yet designing a valid and easy-to-use statistical test for the order is challenging. To date, most results on hypothesis tests have focused on homogeneity, a special case where the null model has order I. In this work, we designed an EM test for the general problem of testing the null hypothesis of order m(0) versus an alternative hypothesis of order larger than m(0). For any positive integer m(0), the null limiting distribution of the EM test is a mixture of chi(2) distributions. The weights in this mixture-limiting distribution can be conveniently computed. Compared with related results, the new result is obtained under much less strict requirements on the component distribution and the parameter space. Extensive simulation studies show that the limiting distributions closely match the finite sample distributions of the EM test. When m(0) = 2, the new EM test has more accurate type I errors and matches the power of the modified likelihood ratio test. When m(0) = 3, there is a clear indication that the test has good power properties. Supplementary materials for this article are available online.