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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=751465
• 关键词: Likelihood; based; Tests; Number; Components; 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.

多元正常分布的有限混合物已广泛用于潜水领域（例如统计遗传学和金融）的经验应用中。在有限的混合模型和制度开关模型中，组件和制度的数量是重要的参数。尽管其重要性，但这些模型中组件和制度的数量的测试一直是一个长期的未解决问题，因为对可能性比率测试（LRT）静态的标准不对称分析由于问题（例如不可识别的参数）而破坏了静态问题，并且真正的参数在参数空间的边界上。在正常混合物和不平等差异的正常混合物中，LRT统计数据的不对称分布仍然是一个悬而未决的问题，因为正常混合物具有其他不可避免的数学特性，可以使现有作品中的关键假设无效，例如“缺乏强大的识别”，如Chen（1995年）（1995年，统计数据），以及无限型渔民的信息。尽管最近有几篇关于LRT主题的论文，涉及单变量有限混合物的正常回归模型中的组件数量，但即使在一个简单的情况下，即使在一个简单的案例中，针对两种组合假设的替代假设，即使在一个简单的案例中，LRT统计量的不对称分布仍然是一个悬而未决的问题。尚未得出Markov制度切换模型中LRT的不对称分布，用于测试M制度的无效假设，当M大于2时。提出的项目由四个子项目组成，每个项目都旨在为基于Migeles的Migeliphatents的基于MIGELISS的基于Mige oferative for oferative for oferative for oferative for oferativatient for oferativatient for oferativatient for oferativatient for My hypothers for My（m+sypose for of for oseform fors fors fors fors for of forsem）（m+）（m+1）有限混合物或制度开关模型，即（i）多变量正常混合物，（ii）混合比例取决于协变量的有限混合模型，（iii）马尔可夫制度切换模型。对于每类模型，我们计划开发一个正交参数化，该参数化提取了Fisher Information矩阵奇异的方向。在不同类别的模型中有所不同的适当重聚化。在提出的重新聚集化下，我们计划表明，对数可能性函数在重新聚集参数的多项式的二次形式上局部近似，从而简单地表征了LRT统计的渐近分布。我们的分析基于LE CAM在二次平均值（DQM）扩展中的可区分版本，该版本在身份丧失下扩展了似然比，在这种情况下，由于Fisher Information Matrix的奇异性，需要高阶扩展。我们还计划提出EM检验，并建立参数引导的渐近有效性。