Generalized canonical analysis.

Abstract

Canonical Analysis is the statistical study of the relationships of two vector variables. In this dissertation we have generalized it to the simultaneous study of the relationships among more than two vectors. We call this extension of Canonical Analysis, Generalized Canonical Analysis (GCA). Besides a detailed exploration of the geometrical properties of GCA, an explicit model and test statistics are developed and it is shown how such model is a generalization of well known univariate and multivariate statistical linear models and methods, such as Canonical Analysis, Multiple Regression Analysis, Univariate and Multivariate Analysis of Variance and Covariance, Discriminant Analysis, Correspondence Analysis and Principal Components Analysis. Other statistical models usually not treated in the literature may still be looked upon as particular cases of the GCA model. Some of these models, which we refer to as Block Regression Models, are treated in more detail. The test statistics developed for this unifying theory of GCA are straightforward and, when used in each of the particular cases, reduce to the test statistics commonly used in the usual linear models. These statistics are generalizations statistics known in other situations. The Maximum Likelihood Estimators (MLE's) for the mean vector and variance-covariance matrix $(\Sigma)$ of a multivariate normal distribution, when different masses are assigned to each observation of a random sample, are obtained. It is shown that the distribution of the MLE of $\Sigma$ is then approximately Wishart. Based on this result, the distributions of the submatrices, and meaningful functions of such submatrices, of the MLE of $\Sigma$ are studied. The results obtained enabled us to investigate the distributions of the above test statistics and their relation to Wilks' lambda. This dissertation also gives an asymptotic normal distribution for the generalized Wilks' lambda. The application of the GCA model and associated test statistics to the study of relationships among categorical variables is also considered and illustrated.