Testing homogeneity in a scale mixture of normal distributions

Abstract

Scale mixtures of normal distributions, i.e., mixtures of normal distributions on the variance parameters, have wide applications. The testing of homogeneity is one of the fundamental challenges in the application of scale mixtures of normal distributions. Recently, Chen and Li (Ann Stat 37:2523–2542, 2009) proposed a class of EM-tests for testing homogeneity in mixtures of normal distributions on the mean parameters and both the mean and variance parameters. In this paper, we retool the EM-test proposed in Chen and Li (2009) for testing homogeneity in scale mixtures of normal distributions. We show that the EM-test has the simple null limiting distribution \(\frac{1}{2} \chi ^2_0 + \frac{1}{2}\chi ^2_1\), an equal mixture of a distribution with point mass at zero and a \(\chi ^2_1\) distribution. We also use a computational method to provide an empirical value for the tuning parameter selection. Simulation studies show that the EM-test has an accurate size and is more powerful than existing methods such as the likelihood ratio test and the method in Chen and Li (2009) when the data are generated from scale mixtures of normal distributions or mixtures of normal distributions on both the mean and variance parameters with two component means being slightly different. To demonstrate the application of the proposed method, we analyze a real-data example.