Testing for the Mixture Hypothesis with Misspecified Distributions in Exponential Family
Jin Seo Cho, Halbert L. White · 2015
We examine the likelihood ratio (LR) statistic testing for the mixture hypothesis when the mixtures of distributions in exponential family are incorrectly specified. The analysis assuming the correct model specification condition in Cho andWhite (2007, Economet-rica) cannot be exactly applied to the misspecified models in a way similar to identified models. Further, the LR statistic often becomes degenerate under the null that the single component based model attains the same likelihood as the mixture. We thus provide a set of regularity conditions for the LR statistic to be non-degenerate asymptotically under the null. Given our regularity conditions, the LR statistic converges weakly to the square of half-normal random variable, non-central chi-square random variable or the maximum of these under the null, depending on the model scopes and data generating processes considered in the text. This is a different consequence from the correctly specified model case, in which the asymptotic null distribution is given as a functional of a Gaussian process.