APPROVAL

The main objective of this thesis is to develop procedures for making inferences about the eigenvalues and eigenvectors of a covariance matrix. Specifically, new procedures for examining dimension reduction in principal component analysis (PCA) are developed. The dimension reduction consists of the following two aspects: reduction in the number of components and reduction in the number of original variables. The procedures are based on a likelihood approach. Parameterizations of eigenvalues and eigenvectors are presented. The parameterizations allow arbitrary eigenvalue multiplicities. The use of the Fisher scoring algorithm for computing maximum likelihood estimates of the covariance parameters subject to multiplicity and other constraints is discussed. Asymptotic distributions of estimators of covariance parameters are derived under normality and non-normality. Likelihood ratio tests and Bartlett corrections are described. Simulation studies show the effectiveness of Bartlett corrections. The new procedures are demonstrated to give better overall results than some existing methods