Likelihood-based tests for the Number of Components in Finite Mixture Models

有限混合模型中分量数量的基于似然的检验

• 批准号: RGPIN-2014-06221
• 负责人: Kasahara, Hiroyuki
• 金额: $ 0.8万
• 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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635814
• 关键词: Likelihood; based; tests; Number; Components

Finite mixtures of normal distributions have been used in numerous empirical applications across various fields such as biological, physical, and social sciences, including finance, economics, and marketing. Mixture-of-expert models with normal component distribution, which have been used in numerous regression, classi?cation, and fusion applications in healthcare, ?nance, surveillance, and recognition, can be viewed as finite mixture of normal regression models. The number of components is an important parameter in applications of finite mixture normal regression models. In economics applications, the number of components often represents the number of unobservable types or abilities. In many other applications, the number of components signifies the number of clusters or latent classes in the data. Despite its importance, testing for the number of components in finite mixture normal regression 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 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. This project studies likelihood-based testing of the null hypothesis of m components against the alternative of (m+1) components in general finite normal mixture models with a vector mixing parameter and a structural parameter, including finite mixture normal regression models with heteroskedastic components. Our project intend to make the following contributions. We develop an orthogonal parameterization that extracts the direction in which the Fisher information matrix is singular. Under this reparameterization, 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. Based on this reparameterization, we derive the asymptotic distribution of the LRT statistic for testing the null hypothesis of m components for m larger than 2 in a mixture model with a multidimensional mixing parameter and a structural parameter. Implementing the LRT has, however, practical difficulties because (i) in some mixture models that are popular in applications (e.g., Weibull duration models and normal mixture models), the Fisher information may not be finite, (ii) the asymptotic distribution depends on the choice of the support of the parameter space, and (iii) simulating the supremum of a Gaussian process is computationally challenging because of the curse of dimensionality.To circumvent these difficulties, We propose a modified EM test by building on this local quadratic representation and extending the EM approach pioneered by Li and Chen (2010, Journal of the American Statistical Association). Given our preliminary results, we expect that the asymptotic null distribution of the proposed modified EM test statistic will be easily simulated. Furthermore, the modified EM test does not suffer from the infinite Fisher information problem.

正常分布的有限混合物已在各个领域的众多经验应用中使用，例如生物，物理和社会科学，包括金融，经济学和营销。具有正常组件分布的特科模型的混合物，这些模型已用于医疗保健中的许多回归，分类和融合应用，可以看作是正常回归模型的有限混合物。组件的数量是有限混合物正常回归模型应用中的重要参数。在经济学应用中，组件的数量通常代表不可观察的类型或能力的数量。在许多其他应用程序中，组件的数量表示数据中集群或潜在类的数量。尽管具有重要性，但对有限混合物中的组件数量的测试正常回归模型一直是一个长期的未解决问题，因为对可能性比率测试（LRT）统计量的标准渐近分析由于不可识别的参数和诸如诸如非可识别参数的问题以及该范围的真实参数而崩溃。在正常混合物和不等差异的正常混合物中，LRT统计数据的渐近分布仍然是一个悬而未决的问题，因为正常混合物具有额外的不良数学特性，可以使现有作品中的关键假设无效，例如“缺乏强大的可识别性”，如Chen（1995年）（1995年，统计学）和无知的Cixs fivection fivection chen（1995年）所讨论的。该项目研究了基于载体混合参数和结构参数的一般有限正常混合模型中的（M+1）组件的替代方案的基于MAMEVENTENS的零假设的可能性测试，其中包括有限混合物与异性疾病分量的有限混合物回归模型。我们的项目打算做出以下贡献。我们开发了一个正交参数化，该参数提取了Fisher信息矩阵奇异的方向。在这种重新聚集化下，对数可能性函数在重新聚集参数的多项式二次形式上局部近似，从而导致了LRT统计的渐近分布的简单表征。基于此重新聚集化，我们得出了LRT统计量的渐近分布，用于测试具有多维混合参数和结构参数的混合模型中M大于2的M分量的零假设。 Implementing the LRT has, however, practical difficulties because (i) in some mixture models that are popular in applications (e.g., Weibull duration models and normal mixture models), the Fisher information may not be finite, (ii) the asymptotic distribution depends on the choice of the support of the parameter space, and (iii) simulating the supremum of a Gaussian process is computationally challenging because of the curse of dimensionality.To避免了这些困难，我们提出了一项修改的EM测试，通过建立这种局部二次表示，并扩展了Li and Chen（2010年，美国统计协会杂志）开创的EM方法。鉴于我们的初步结果，我们希望将很容易模拟所提出的修改EM检验统计量的渐近无效分布。此外，经过修改的EM测试不会遭受无限的Fisher信息问题。