A moment-based test for the homogeneity in mixture natural exponential family with quadratic variance functions

In this paper, we propose a simple moment-based procedure for testing homogeneity in the natural exponential family with quadratic variance functions. In the literature, solutions to this problem normally involve establishing identifiability of parameters first, then testing the hypotheses whether the data come from a single distribution or a mixture of distributions. Our procedure directly tests the hypotheses without the need to establish parameter estimability. Simulation studies demonstrate that the power of our test is comparable to the supplementary score test and the separate score test proposed by Wu and Gupta [Wu, Y., Gupta, A.K., 2003. Local score tests in mixture exponential family. J. Statist. Plann. Inference 116, 421–435], and the normalized score test proposed by Shoukri and Lathrop [Shoukri, M., Lathrop, G.M., 1993. Statistical testing of genetic linkage under heterogeneity. Biometrics 49, 151–161]. In these simulation studies, the methods by the others are specific to cases with a known null distribution, and our methods can also be applied to cases with unknown null distribution. Our test procedure is demonstrated on two real data sets.