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Statistica Sinica 19 (2009), 1603-1619



Yuejiao Fu, Jiahua Chen and John D. Kalbfleisch

York University, University of British Columbia and University of Michigan

Abstract: We consider testing for homogeneity in a two-sample problem in which one of the samples has a mixture structure. The problem arises naturally in many applications such as case-control studies with contaminated controls, or the test of a treatment effect in the presence of nonresponders in biological experiments or clinical trials. In this paper, we suggest using the modified likelihood ratio test (MLRT), which is devised to restore a degree of regularity in the mixture situation. The asymptotic properties of the MLRT statistic are investigated in mixtures of general one-parameter kernels, and in a situation where the kernels have an additional structural parameter. The MLRT statistic is shown to have a simple $\chi_1^2$ null limiting distribution in both cases and simulations indicate that the MLRT performs better than other tests under a variety of model specifications. The proposed method is also illustrated in an example arising from a trial relating to morphine addiction in rats.

Key words and phrases: Asymptotic distribution, likelihood ratio test, mixture models, normal mixture, structural parameter, two-sample problem.

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