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

正态分布的有限混合已被用于各种领域的大量经验应用，如生物、物理和社会科学，包括金融、经济和市场营销。具有正态成分分布的专家混合模型可以看作是正态回归模型的有限混合，这些模型在医疗保健、护理、监测和识别等领域中得到了大量的回归、分类和融合应用。分量个数是有限混合正态回归模型应用中的一个重要参数。在经济学应用中，组件的数量通常代表无法观察到的类型或能力的数量。在许多其他应用中，组件的数量表示数据中的簇或潜在类的数量。**尽管有限混合正态回归模型的分量个数检验很重要，但由于似然比检验(LRT)统计量的标准渐近分析由于参数空间边界上的不可识别参数和真实参数等问题而失效，因此对有限混合正态回归模型的分量个数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。**本课题研究具有向量混合参数和结构参数的一般有限正态混合模型(包括具有异方差分量的有限混合正态回归模型)中m个分量的零假设相对于(m+1)个分量备选的基于似然的检验。我们的项目打算做出以下贡献。**我们开发了一种正交参数化法，它提取Fisher信息矩阵的奇异方向。在这种重新参数化下，对数似然函数被重新参数化参数的二次型多项式局部逼近，从而简单地刻画了LRT统计量的渐近分布。在此基础上，我们给出了具有多维混合参数和结构参数的混合模型中检验m>2的m个分量的零假设的LRT统计量的渐近分布。**然而，LRT的实现有实际困难，因为(I)在一些应用广泛的混合模型(例如，威布尔久期模型和正态混合模型)中，Fisher信息可能不是有限的，(Ii)渐近分布取决于参数空间的支撑性的选择，以及(Iii)由于维度灾难，模拟高斯过程的上确界在计算上具有挑战性。**为了避免这些困难，我们提出了一种改进的EM检验，它建立在局部二次表示的基础上，并扩展了Li和Chen(2010，Journal of the American Statistics Association)开创的EM方法。给出我们的初步结果，我们预计所提出的修正EM检验统计量的渐近零分布将很容易被模拟。此外，改进的EM检验不受无限Fisher信息问题的影响。

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.