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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=713400
• 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已广泛应用于统计遗传学和金融等不同领域的实证应用中。在有限混合模型和状态切换模型中，组件和状态的数量是一个重要参数。尽管很重要，但对这些模型中的组件和状态的数量进行测试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会因参数不可识别和真实参数位于参数空间边界等问题而失败。在方差不等的正态混合物中，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良数学特性，该特性使现有工作中的关键假设无效，例如 Chen（1995 年，统计年鉴）所讨论的“缺乏强可识别性”以及关于混合比例的无限费舍尔信息。 虽然最近有一些关于单变量有限混合正态回归模型中分量数量的 LRT 主题的论文，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，即使在测试一个分量的原假设与两个分量的备择假设的简单情况下也是如此。当 M 大于 2 时，尚未导出马尔可夫状态切换模型中状态数量的 LRTS 渐近分布来检验 M 状态的原假设。 拟议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，对 M 分量的零假设与在一类有限混合或状态切换模型中 M 大于 2 的 (M+1) 分量的备择假设进行检验，即 (i) 多元正态混合物，(ii) 有限混合模型，其中混合比例取决于协变量，(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来提取费舍尔信息矩阵奇异的方向。不同类别模型之间存在差异的适当重新参数化。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化参数的多项式的二次形式局部近似的，从而对 LRT 统计量的渐近分布进行简单的表征。 我们的分析基于 Le Cam 的二次均值可微分 (DQM) 展开的一个版本，该展开在可识别性损失的情况下展开似然比，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 检验并建立参数引导的渐近有效性。