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在统计遗传学、金融等领域有着广泛的应用，在有限混合模型和状态切换模型中，分量个数和状态个数是一个重要的参数。尽管很重要，但对这些模型中的组件和状态的数量进行检验一直是一个长期未解决的问题，因为似然比检验(LRT)统计量的标准渐近分析由于参数不可识别和真实参数处于参数空间边界等问题而崩溃。在具有不等方差的正态混合中，LRT统计量的渐近分布仍然是一个悬而未决的问题，因为正态混合有一个额外的不受欢迎的数学性质，这使得现有工作中的关键假设无效，如Chen(1995，Annals of Statistics)所讨论的“缺乏强可辨识性”和关于混合比例的无限Fisher信息。虽然最近已经有一些关于一元有限混合正态回归模型中分量个数的LRT问题的文章，但即使在检验一个分量的零假设与两个分量的替代假设的简单情况下，多元正态混合的LRT统计量的渐近分布仍然是一个悬而未决的问题。当M大于2时，马尔可夫状态转换模型中状态个数的LRT的渐近分布还没有被推导出来用于检验M个状态的零假设。该项目由四个子项目组成，每个子项目的目的是针对一类有限混合或状态转换模型中的(M+1)个分量大于2的备选假设，即(I)多变量正态混合，(Ii)混合比例依赖于协变量的有限混合模型，(Iii)马尔可夫状态转换模型中的(M+1)个分量的备选假设，发展一个基于似然检验的检验。对于每一类模型，我们计划开发一种正交参数化，以提取Fisher信息矩阵的奇异方向。适当的重新参数化，在不同的模型类中有所不同。在所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的多项式的二次型局部逼近的，从而简单地刻画了LRT统计量的渐近分布。我们的分析基于Le Cam的二次平均可微(DQM)展开式，该展开式在失去可辨识性的情况下扩展了似然比，其中由于Fisher信息矩阵的奇异性而需要更高阶的展开式。我们还计划提出EM检验以及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.