Mathematical Sciences: Algebraic Methods in Multivariate Statistical Analysis

建议继续研究多元正态模型的统计特性，这些模型由协方差矩阵上的格条件独立（LCI）假设和/或群对称假设确定，并通过对均值的相容线性限制来增强。LCI模型已被证明适用于分析非嵌套缺失数据模式的多元正态数据集。LCI模型的一个新的应用将在这里强调：他们的应用程序的非嵌套依赖线性回归模型，在计量经济学中称为看似无关回归（SUR）模型的分析。SUR模型可以被认为是一个有限的非嵌套的线性回归子空间的集合，在回归之间具有相关的误差。对于给定的SUR模型，可以确定与平均结构相容的最小限制性LCI协方差模型，从而得到SUR模型的显式最大似然估计。到目前为止，LCI模型只研究了多元正态分布。该建议的另一个新方面是LCI模型在多向列联表中的分类数据的应用。在正态数据的情况下，这样的LCI模型应该允许显式的最大似然估计列联表与非嵌套的缺失类别。在经典的多变量分析（相关数据的研究）中出现的许多熟悉的统计模型可以被视为根据均值和/或协方差的自然代数条件定义的模型的特殊情况。这种观点将（a）导致这些模型的统一和明确的（非迭代）分析，（B）通过允许经典方法应用于许多新模型，以及允许在非嵌套模式和非嵌套回归子空间中发生缺失数据的可能性，扩展多变量分析的范围。

It is proposed to continue the study of the statistical properties of multivariate normal models determined by lattice conditional independence (LCI) assumptions and/or group symmetry assumptions on the covariance matrix, augmented by compatible linear restrictions on the mean. LCI models have been shown to be applicable to the analysis of multivariate normal data sets with nonnested missing data patterns. A new application of LCI models will be emphasized here: their application to the analysis of a collection of nonnested dependent linear regression models, known in econometrics as a seemingly unrelated regression (SUR) model. A SUR model may be thought of as a finite nonnested collection of linear regression subspaces with correlated errors across regressions. For a given SUR model, the least restrictive LCI covariance model compatible with the mean structure can be determined, leading to explicit maximum likelihood estimates for the SUR model. To date, LCI models have been studied only for multivariate normal distributions. Another new aspect of this proposal is the application of LCI models to categorical data in multiway contingency tables. As in the case of normal data, such LCI models should allow explicit maximum likelihood estimates for contingency tables with nonnested missing categories. Many familiar statistical models occurring in classical multivariate analysis (the study of correlated data) can be viewed as special cases of models defined in terms of natural algebraic conditions on the means and/or covariances. This viewpoint will (a) lead to a unified and explicit (non-iterative) analysis of these models, and (b) expand the scope of multivariate analysis by allowing the application of classical methods to many new models, as well as allowing the possibilities of missing data occurring in nonnested patterns and of nonnested regression subspaces.