Testing homogeneity in a heteroscedastic contaminated normal mixture

Large-scale simultaneous hypothesis testing appears in many areas. A well-known inference method is to control the false discovery rate. One popular approach is to model the z-scores derived from the individual t-tests and then use this model to control the false discovery rate. We propose a heteroscedastic contaminated normal mixture to describe the distribution of z-scores and design an EM-test for testing homogeneity in this class of mixture models. The proposed EM-test can be used to investigate whether a collection of z-scores has arisen from a single normal distribution or whether a heteroscedastic contaminated normal mixture is more appropriate. We show that the EM-test statistic has a shifted mixture of chi-squared limiting distribution. Simulation results show that the proposed testing procedure has accurate type-I error and significantly larger power than its competitors under a variety of model specifications. A real-data example is analysed to exemplify the application of the proposed method.