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

正态分布的有限混合分布已经在各个领域的许多经验应用中使用，例如生物，物理和社会科学，包括金融，经济和营销。 混合专家模型与正常成分分布，已被用于许多回归，classi？阳离子和融合在医疗保健中的应用，Nance、Surveillance和Recognition可以看作是正态回归模型的有限混合。 在有限混合正态回归模型的应用中，分量个数是一个重要的参数。在经济学应用中，成分的数量通常代表不可观察的类型或能力的数量。在许多其他应用中，组件的数量表示数据中聚类或潜在类的数量。 * * 尽管它的重要性，测试有限混合正态回归模型中的分量数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由于诸如不可识别的参数和参数空间边界上的真参数等问题而失败。 在具有不等方差的正态混合物中，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期望的数学性质，使现有工作中的关键假设无效，例如Chen（1995，Annals of Statistics）讨论的“缺乏强可识别性”和关于混合比例的无限Fisher信息。 * * 本项目研究在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正态混合回归模型）中，m个分量的零假设相对于（m+1）个分量的备择假设的基于似然的检验。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，提取Fisher信息矩阵奇异的方向。在这种重新参数化，对数似然函数是局部近似的二次多项式形式的重新参数化的参数，导致一个简单的表征的LRT统计量的渐近分布。基于这种重新参数化，我们得到了渐近分布的LRT统计检验零假设的m个组件，m大于2的混合模型中的多维混合参数和结构参数。** 然而，实施LRT具有实际困难，因为（i）在应用中流行的一些混合模型（例如，Weibull持续时间模型和正态混合模型），Fisher信息可能不是有限的，（ii）渐近分布取决于参数空间支持的选择，以及（iii）由于维数灾难，模拟高斯过程的上确界在计算上具有挑战性。为了克服这些困难，我们提出了一种改进的EM测试，通过建立在这种局部二次表示的基础上，并扩展了Li和Chen开创的EM方法（2010，Journal of the American Statistical Association）。鉴于我们的初步结果，我们预计，所提出的修改后的EM检验统计量的渐近零分布将很容易模拟。此外，修改的EM测试不受无限Fisher信息问题。

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.