Test for homogeneity in gamma mixture models using likelihood ratio

A testing problem of homogeneity in gamma mixture models is studied. It is found that there is a proportion of the penalized likelihood ratio test statistic that degenerates to zero. The limiting distribution of this statistic is found to be the chi-bar-square distributions. The degeneration is due to the negative-definiteness of a complicated random matrix, depending on the shape parameter under the null hypothesis. In light of this dependency, bounds on the distribution are introduced and a weighted average procedure is proposed. Simulation suggests that the results are accurate and consistent, and that the asymptotic result applies to the maximum likelihood estimator, obtained via an Expectation-Maximization algorithm.