Algebraic Methods in Multivariate Statistical Analysis

建议继续研究由协方差矩阵上的格条件独立（LCI）假设和/或群对称假设决定的多元正态模型的统计性质，并通过对平均值的相容线性限制进行扩充。LCI模型已被证明适用于分析具有非嵌套缺失数据模式的多变量正态数据集。这里将强调LCI模型的一个新应用：它们用于分析一组非嵌套相关线性回归模型，在计量经济学中称为看似不相关的回归（SUR）模型。SUR模型可以被认为是线性回归子空间的有限非嵌套集合，这些子空间具有跨回归的相关误差。对于给定的SUR模型，可以确定与平均结构兼容的最小约束LCI协方差模型，从而得到SUR模型的显式最大似然估计。迄今为止，LCI模型只针对多元正态分布进行了研究。该建议的另一个新方面是将LCI模型应用于多路列联表中的分类数据。与正常数据的情况一样，这种LCI模型应该允许对具有非嵌套缺失类别的列联表进行显式的最大似然估计。在经典多变量分析（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)导致对这些模型进行统一和明确的（非迭代的）分析，并且(b)通过允许将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出现丢失数据的可能性，扩展多元分析的范围。

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.