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分布在统计遗传学和金融学等不同领域的经验应用中得到了广泛的应用。在有限混合模型和状态转换模型中，组分和状态的数目是一个重要的参数。尽管它的重要性，测试这些模型中的组件和制度的数量一直是一个长期未解决的问题，因为标准的渐近分析的似然比检验（LRT）统计故障，由于诸如不可识别的参数和真正的参数是在参数空间的边界上的问题。在具有不等方差的正态混合物中，LRT统计量的渐近分布仍然是一个悬而未决的问题，因为正态混合物具有额外的不期望的数学性质，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”和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分量数的LRT的，但多变量正态混合的LRT统计量的渐近分布仍然是一个悬而未决的问题，即使在一个简单的情况下，检验一个分量的零假设对两个分量的备择假设。马尔可夫状态转换模型中状态数的LRTS的渐近分布尚未被导出，用于检验M个状态的零假设，当M大于2时。** 拟议项目由四个子项目组成，每个子项目的目的是在一类有限混合或状态转换模型中，针对M大于2的（M+1）个分量的备择假设，对M个分量的零假设进行似然检验，即（i）* 多元正态混合，（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。** 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。一个适当的重新参数化，在不同的模型类别中有所不同。根据建议的重新参数化，我们计划表明，对数似然函数是局部近似的二次多项式形式的重新参数化的参数，导致一个简单的表征的渐近分布的LRT统计。** 我们的分析是基于一个版本的Le Cam的二次均值可微（DQM）扩展，该扩展在可识别性损失下的似然比，其中由于Fisher信息矩阵的奇异性而需要更高阶的扩展。我们还计划提出EM检验以及建立参数自助的渐近有效性。

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.