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.
