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

有限混合和马尔可夫政权切换模型中组件/政权数量的基于似然的检验

• 批准号: RGPIN-2019-04047
• 负责人: Kasahara, Hiroyuki
• 金额: $ 1.22万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675812
• 关键词: Likelihood; based; Tests; Number; Components

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.

多元正态分布的有限混合在统计遗传学和金融等不同领域的实证应用中有着广泛的应用。在有限混合模型和区域转换模型中，组分和区域的数目是一个重要的参数。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实参数处于参数空间的边界等问题而失效，因此对这些模型中的分量和区域的数量的检验一直是一个长期未解决的问题。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，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也是如此。当M大于2时，马尔可夫状态转换模型中状态数目的LRT的渐近分布还没有被推导出来用于检验M状态的零假设。*所提出的项目由四个子项目组成，每个子项目旨在针对一类有限混合或状态转换模型中M的(M+1)个分量大于2的备选假设，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，开发基于似然的检验。(3)马尔可夫状态转换模型。*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。*我们的分析是基于Le Cam的二次平均可微(DQM)展开式，该展开式在失去可识别性的情况下扩展了似然比，其中由于Fisher信息矩阵的奇异性，需要进行高阶展开式。我们还计划提出EM检验以及建立参数自举的渐近有效性。