Estimating the dimensionality in canonical correlation analysis

Statisticians are often faced with the problem of choosing the appropriate dimensionality of a model that will fit a given set of observations. In canonical correlation analysis, the number of nonzero population canonical correlation coefficients is called the dimensionality. Several methods are examined for estimating the dimensionality in canonical correlation analysis. A likelihood ratio test (LRT) procedure is often used for testing a sequence of dimensionality hypotheses. A second method is based on maximizing the marginal (log) likelihood for the dimensionality. Another is based on Akaike's (1973) information criterion for choice of models. Akaike's information criterion can be extended to make it consistent and this extended criterion is similar to Schwarz's (1978) Bayesian criterion. And another method uses Mallows's (1973) {dollar}Csb{lcub}p{rcub}{dollar} statistic for selection of variables in regression.; Various statistical properties of these estimation methods will be examined. The LRT procedure tests hypotheses sequentially giving rise to the question of how the size of the test is affected by viewing the sequence of tests as conditional tests. It is shown that for normal populations the size of the test is not affected asymptotically. This result is robust to certain types of departures from normality. The asymptotic distribution of the dimensionality estimated by the marginal (log) likelihood method is developed for normal populations. Fujikoshi (1985) studied the statistical properties of Akaike's and Mallows's method for canonical correlation analysis for normal populations. These results are shown to be sensitive to departures from multivariate normality. The asymptotic distributions of the dimensionalities estimated by Mallows's criterion and Akaike's information criterion are given for (non-normal) multivariate populations with finite fourth moments. These distributions have a simplified form in the case of elliptical populations, and modified criteria are proposed. Monte Carlo results will also be given, including an overall comparison of the various methods.