Testing for a Finite Mixture Model with Two Components

We consider a finite mixture model with k components and a kernel distribution from a general parametric family. We consider the problem of testing the hypothesis k = 2 against k ≥ 3. In this problem, the likelihood ratio test has a very complicated large sample theory and is difficult to use in practice. We propose a test based on the likelihood ratio statistic where the estimates of the parameters, (under the null and the alternative) are obtained from a penalized likelihood which guarantees consistent estimation of the support points. The asymptotic null distribution of the corresponding modified likelihood ratio test is derived and found to be relatively simple in nature and easily applied. Simulations based on a mixture model with normal kernel are encouraging that the modified test performs well, and its use is illustrated in an example involving data from a medical study where the hypothesis arises as a consequence of a potential genetic mechanism.