Testing Homogeneity in a Semiparametric Two-Sample Problem

We study a two-sample homogeneity testing problem, in which one sample comes from a population with density 𠑓 ( 𠑥 ) and the other is from a mixture population with mixture density ( 1 − 𠜆 ) 𠑓 ( 𠑥 ) + 𠜆 𠑔 ( 𠑥 ) . This problem arises naturally from many statistical applications such as test for partial differential gene expression in microarray study or genetic studies for gene mutation. Under the semiparametric assumption 𠑔 ( 𠑥 ) = 𠑓 ( 𠑥 ) 𠑒 𠛼 + 𠛽 𠑥 , a penalized empirical likelihood ratio test could be constructed, but its implementation is hindered by the fact that there is neither feasible algorithm for computing the test statistic nor available research results on its theoretical properties. To circumvent these difficulties, we propose an EM test based on the penalized empirical likelihood. We prove that the EM test has a simple chi-square limiting distribution, and we also demonstrate its competitive testing performances by simulations. A real-data example is used to illustrate the proposed methodology.