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

多元正态分布的有限混合已广泛应用于统计遗传学和金融等多个领域的实证应用。在有限混合模型和状态切换模型中，组件数和状态数是一个重要的参数。尽管它很重要，但由于似然比检验（LRT）统计量的标准渐近分析由于参数不可识别和真参数位于参数空间边界等问题而失效，因此对这些模型中成分和制度的数量的测试一直是一个长期未解决的问题。在方差不等的正态混合中，LRT统计量的渐近分布仍然是一个悬而未决的问题，因为正态混合具有额外的不良数学性质，使现有工作中的关键假设无效，例如Chen （1995, Annals of Statistics）讨论的“缺乏强可识别性”以及关于混合比例的无限Fisher信息。虽然最近有几篇论文是关于单变量有限混合正态回归模型中成分数的LRT的主题，但多元正态混合的LRT统计量的渐近分布仍然是一个开放的问题，即使在一个简单的情况下，检验一个成分的零假设对两个成分的备用假设。当M大于2时，为了检验M状态的零假设，还没有得到马尔可夫状态切换模型中状态数的LRTS的渐近分布。提议的项目由四个子项目组成，每个子项目旨在开发基于似然的检验M个成分的零假设对M大于2的（M+1）个成分的备选假设在一类有限混合或状态切换模型中，即(i)多元正态混合，（ii）混合比例取决于协变量的有限混合模型，（iii）马尔可夫状态切换模型。对于每一类模型，我们计划开发一个正交参数化，提取Fisher信息矩阵奇异的方向。在不同类型的模型中不同的适当的重新参数化。在提出的重新参数化下，我们计划证明对数似然函数是由重新参数化参数的多项式的二次形式局部逼近的，从而导致LRT统计量的渐近分布的简单表征。我们的分析是基于Le Cam的二次均值可微（DQM）展开的一个版本，该版本在可识别性丧失的情况下扩展了似然比，其中由于Fisher信息矩阵的奇异性需要更高阶的展开。我们还计划提出EM检验，并建立参数自举的渐近有效性。

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.