Distributions Of Latent Roots Research Articles

Distribution of the residual roots in principal components analysis.

The latent of distribution of latent roots of the covariance martix of normal variables, when a hypothetical linear function of the variables is eliminated, is derived in this paper. The relation between original roots and the residual roots- after elimination of, is also derived by an analytical method. An exact test for the goodness of fit of a single nonisotropic hypothetical principal components, using the residual roots, is then obtained.

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

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

Recurrence relations of coefficients of the generalized hypergeometric function in multivariate analysis

James(1960) defined the zonal polynomials and used it to represent the joint distributions of latent roots of VVisfiart matrix. The zonal polviionnals played an important role to define the generalized hypergeometric function of symmetric matrix argument Since then, many density functions and moments based on Wishart matrix have been expressed in terms of the generalized hy¬pergeometric Function. The purpose of this paper is to get the recurrence relations for the coefficients of it. In Section 1 we derive a partial differen¬tial equations having the generalized hypergeometric function as the unique solution. Then we ubtain the recurrence relations until order 7 in Section 2.

Invariant Polynomials and Related Tests

SYNOPTIC ABSTRACTMany test statistics for testing hypotheses on the parameters of one or more multi-normal populations are associated with the latent roots of certain positive definite symmetric matrices. Thus the distributions of latent roots play vital roles in some testing problems. One such case is the joint distribution of the latent roots of a Wishart matrix. One representation of this joint density is in terms of zonal polynomials. Zonal polynomials are certain homogeneous polynomials in the latent roots of a symmetric matrix, which are invariant under orthogonal transformations. Such invariant polynomials and related tests are examined in this article. Some applications of these topics such as multivariate calibration, regression models, discriminant analysis, robustness studies and econometric theory are also considered.

On the distributions of the latent roots and traces of certain random matrices

It is shown that differential equations given by the author may be used recursively to construct certain multivariate null distributions in reduced form. These include the distributions of individual latent roots of B = S 1(S 1 + S 2) −1 , and distributions of Tr B and Tr S 1S 2 −1 , for small numbers of variates.