Mathematical Sciences: Multivariate Statistical Analysis

本研究涉及：(1)多元正态分布中条件独立性的格型模型；(2)均值向量和协方差矩阵都有格型约束的正态模型；(3)非嵌套缺失数据模型；(4)群对称协方差模型；(5)格型条件独立性约束和群对称条件相结合的协方差模型；(6)刻画广义下三角矩阵群的传递作用；(7)正定矩阵行列式的Hadamard不等式的推广；(8)不变多元检验的相合性；(9)独立Gamma随机变量加权和的尾概率的界；以及(10)组合独立显著性检验--对Fisher组合过程的辩护。这项研究属于统计学和概率学的一般领域。这些问题通过代数和分析技术的组合来解决，包括群表示理论、分配格论和不变(Haar)度量，以表征模型的结构，并确定估计量和检验特征的分布和决策性质。

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.