The Efficiency of the Asymptotic Expansion of the Distribution of the Canonical Vector under Nonnormality

Abstract

In canonical correlation analysis, canonical vectors are used in the interpretation of the canonical variables. We are interested in the asymptotic representation of the expectation, the variance and the distribution of the canonical vector. In this study, we derive the asymptotic distribution of the canonical vector under nonnormality. To obtain the asymptotic expansion of the canonical vector, we use a perturbation method. In addition, as an example, we show the asymptotic distribution with an elliptical population. In multivariate statistical analysis, the distributions of latent roots and latent vectors of certain symmetric matrices constructed from the sample covariance matrix are important in some cases and have been studied by many authors. These studies can be used as the basis for the canonical correlation analysis, which is an approach that characterizes the correlation structure between two sets of variables. Considering the distribution of the canonical correlation with the assumption of the multivariate normal population, asymptotic expansions of the distributions were studied by Sugiura (1976), Fujikoshi (1977, 1978), Muirhead (1978) and others. The distributions of a function of latent roots of the sample covariance matrix in nonnormal populations were studied by Fujikoshi (1980), Muirhead and Waternaux (1980), Fang and Krishinaiah (1982), Siotani et al. (1985), Seo et al. (1994) and others. The distribution of the canonical vector was studied by Eaton and Tyler (1994), Boik (1998), Anderson (1999), Taskinen et al. (2006) and others. This paper deals with the asymptotic expansion of the canonical vector under nonnormality. Let us denote x =( x � 1, x � 2) � as p + q dimensional variables with mean µ and