Inference of Latent Roots Using Asymptotic Expansions of Likelihood Functions.

摘要

The most commonly used multivariate techniques include principal component analysis, factor analysis, canonical correlation analysis, multivariate analysis of variance and discriminant analysis. All of these techniques involve the latent roots and vectors of random matrices and it is important to be able to make inferences about the population parameters, or a subset of them. Unfortunately, even under the usual assumption that the observations have been drawn from a multivariate normal population, the exact sampling distributions of these random roots and vectors are generally extremely complicated and it appears difficult, if not impossible, to use them (or the corresponding likelihood functions) directly for inferential purposes. There are two reasons; firstly, the distributions are complicated and very difficult to evaluate numerically (and, as a consequence, the likelihood functions are intractable) and secondly, it is not at all obvious how one should draw inferences about, for example, a few parameters in the presence of many nuisance parameters. This report is concerned with problems such as these in connection with the latent roots occurring in the following three multivariate situations based on normal populations: (1) Principal component analysis; (2) Multivariate analysis of variance and discriminant analysis; and (3) Canonical correlation analysis. (Author)