Likelihood-based Tests for the Number of Components/Regimes in Finite Mixture and Markov Regime Switching Models

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Finite mixtures of multivariate normal distributions have been widely used in empirical applications in diverse fields such as statistical genetics and finance. In finite mixture models and regime-switching models, the number of components and regimes is an important parameter. Despite its importance,  testing for the number of components and regimes in these models has been a long-standing unsolved problem because the standard asymptotic analysis of the likelihood ratio test (LRT) statistic breaks down due to problems such as non-identifiable parameters and the true parameter being on the boundary of the parameter space. In normal mixtures with unequal variances, the asymptotic distribution of the LRT statistic remains an open question because normal mixtures have an additional undesirable mathematical property that invalidates key assumptions in the existing works, such as ``the lack of strong identifiability'' as discussed by Chen (1995, Annals of Statistics) and the infinite Fisher information with respect to mixing proportion. While a few recent papers have been written on the subject of the LRT for the number of components in univariate finite mixture normal regression models, the asymptotic distribution of the LRT statistic for multivariate normal mixtures remains an open question even in a simple case of testing the null hypothesis of one component against the alternative hypothesis of two components. The asymptotic distribution of the LRTS for the number of regimes in Markov regime switching models has not been derived for testing the null hypothesis of M regimes when M is larger than 2. The proposed project composes of four sub-projects, each of which aims at developing a likelihood-based test of the null hypothesis of M components against the alternative hypothesis of (M+1) components for M being larger than 2 in a class of finite mixture or regime switching models, namely, (i) multivariate normal mixtures, (ii) finite mixture models in which mixing proportion depends on covariates, (iii)   Markov regime switching models. For each class of models, we plan to develop an orthogonal parameterization that extracts the direction in which the Fisher information matrix is singular. An appropriate reparameterization that differs across different classes of model. Under the proposed reparameterization, we plan to show that the log-likelihood function is locally approximated by a quadratic form of polynomials of the reparameterized parameters, leading to a simple characterization of the asymptotic distribution of the LRT statistic. Our analysis is based on a version of Le Cam's differentiable in quadratic mean (DQM) expansion that expands the likelihood ratio under the loss of identifiability, where a higher order expansion is required due to the singularity of Fisher information matrix. We also plan to propose the EM test as well as establish the asymptotic validity of the parametric bootstrap.