Recent Developments in Likelihood Theory with Applications to
Testing Homogeneity in Finite Mixture Models and other Models

Yongzhao Shao
Division of Biostatistics
New York University School of Medicine


        The likelihood ratio tests (LRT) are well known to be powerful and optimal judged by various asymptotic relative efficiencies (AREs) including Bahadur's ARE and Hodges-Lehmann ARE.  Importantly, there exist many statistical models which are "non-regular" in the classical sense; however, they are increasingly used in many fields including biology, economics, and genetics. Examples of non-regular models include finite mixture models, change-point models, segmented regression models
and others.  The irregularities that make the likelihood analysis complicated for these models include degeneracy of the Fisher
information matrix, loss of identifiability for parameters in null models, and true parameters being on the boundary of the parameter space, etc. Consequences of these complications include that the likelihood ratio test statistic may not have the typical chi-squared type limit distribution and may even diverge under the null hypothesis.

        In this talk, we review various alternative tests and modifications of the LRT that have been proposed in the literature which are often less powerful but easier to use in practice than the classical LRT. We further demonstrate that some selected variant of the LRT can have higher power than the classical LRT in these non-regular models in both small and large samples.