Likelihood Ratio Testing for a Class of Nonidentifiable Problems

Chong-Zhi Di, Fred Hutchinson Cancer Research Center
Kung-Yee Liang, Johns Hopkins University

Abstract

We consider hypothesis testing problems where some nuisance parameters are present only under the alternative hypothesis. Examples include testing the existence of a change point, testing linearity versus a nonlinear trend, testing the number of components in finite mixture models, and testing linkage under heterogeneity in genetic epidemiology. It is known that the likelihood ratio test (LRT) statistic does not follow a conventional 2 distribution in these problems, due to non-identifiability under the null. In this paper, we consider a general class of nonidentifiable problems and investigate the asymptotic behavior of the LRT. We show that its limiting distribution is equivalent to the supremum of a squared Gaussian process. We also extend the modified/penalized likelihood ratio test (PLRT), proposed by Chen et al. (2001) in finite mixture models, to this general class. It can be shown that the PLRT converges to a 2 distribution under the null, and thus PLRT is convenient to use in practice. The LRT is an omibus test against all types of alternative hypotheses, while the PLRT is powerful against certain types of alternatives but not others. Simulation studies were conducted to evaluate the finite sample performance of the LRT and PLRT.