Mathematical Sciences: Multivariate Statistical Analysis

本研究涉及的统计推断有：(1)多元正态分布中条件独立性的格点模型；(2)对均值向量和协方差矩阵都有格约束的正态模型；(3)非嵌套缺失数据模型；(4)群对称协方差模型；(5)结合格条件独立约束和群对称条件的协方差模型；(6)刻画广义下三角矩阵群的传递作用；(7)关于正定矩阵行列式的Hadamard不等式的推广；(8)不变多变量检验的一致性；(9)独立伽马随机变量加权和尾概率的界；（10）组合独立显著性检验——对Fisher组合法的辩护。这项研究是在统计学和概率论的一般领域。这些问题是由代数和解析技术的结合来解决的，包括群表示理论、分配格理论和不变（Haar）测度，以表征模型的结构，确定估计量的分布和决策理论性质，并测试特征。

This research concerns statistical inference for: (1) Lattice models for conditional independence in a multivariate normal distribution; (2) normal models with lattice restrictions on both the mean vector and covariance matrix; (3) non-nested missing data models; (4) group symmetry covariance models; (5) covariance models combining lattice conditional independence restrictions and group symmetry conditions; (6) characterizing transitive actions of generalized lower triangular matrix groups; (7) extensions of Hadamard's inequality for the determinant of a positive definite matrix; (8) consistency of invariant multivariant tests; (9) bounds for tail probabilities of weighted sums of independent gamma random variables; and (10) combining independent significance tests - a defense of Fisher's combination procedure. This research is in the general area of statistics and probability. The problems are attacked by a combination of algebraic and analytic techniques, including the theory of group representations, distributive lattice theory, and invariant (Haar) measures, in order to characterize the structure of the models and to determine distributional and decision-theoretic properties of estimators and test characteristics.