## Electronic Theses and Dissertations

### Multivariate moments and Cochran theorems.

1993

Doctoral Thesis

Ph.D.

#### Department

Mathematics and Statistics

Wong, Chi Song,

Statistics.

#### Rights

info:eu-repo/semantics/openAccess

This thesis is divided into two related parts: (I) Moments. For a multivariate elliptically contoured random matrix $Y\sim MEC\sb{n\times p}(\mu,\ \Sigma\sb{Y},\ \phi),$ formulae for finding the higher order moments of both Y and its quadratic forms are obtained in terms of $\mu,\ \Sigma\sb{Y}$ and $\phi,$ where $\Sigma\sb{Y}$ is not required to have the form $A\otimes\Sigma.$ These results are so general that they are new even for the normal setting. Specific worked out examples on moments are given for both normal and certain non-normal settings such as multivariate uniform distributions, symmetric multivariate Pearson Type VII distributions, generalized Wishart distributions, multivariate components of variance models and MANOVA models. The proofs involve linear operators in inner product spaces, Kronecker products, multilinear differential forms and adjoint operators of the linear functions. (II) Cochran theorems. For a family of quadratic forms, $\{Q\sb{i}(Y)\}\sbsp{i=1}{\ell},$ of Y with $Q\sb{i}(Y)=Y\sp\prime W\sb{i}Y+B\sbsp{i}{\prime}Y+Y\sp\prime C\sb{i}+D\sb{i},\ W\sb{i}$ symmetric and $Y\sim N\sb{n\times p}(\mu,\ \Sigma\sb{Y}),$ necessary and sufficient conditions are obtained under which $\{Q\sb{i}(Y)\}$ is an independent family of Wishart $W\sb{p}(m\sb{i},\Sigma,\lambda\sb{i})$ random matrices, (*). Such a result is referred to as a Cochran theorem. The Cochran theorems just mentioned are general in that the covariance matrix $\Sigma\sb{Y}$ need not take the form $A\otimes\Sigma$ and need not be positive definite. Some of these results are extended further to the case where either (i) $W\sb{p}(m\sb{i},\Sigma,\lambda\sb{i})$ in (*) is replaced by $DW\sb{p} (m\sb{1i},m\sb{2i},\Sigma,\lambda\sb{1i},\lambda\sb{2i}),$ the distribution of the difference of two independent Wishart random matrices $Q\sb{1i}$ and $Q\sb{2i}$ with $Q\sb{1i}\sim W\sb{p}(m\sb{1i},\Sigma,\lambda\sb{1i})$ and $Q\sb{2i}\sim W\sb{p}(m\sb{2i},\Sigma,\lambda\sb{2i}),$ or (ii) Y is multivariate elliptically contoured distributed. The proofs involve linear operators in inner product spaces, Moore-Penrose inverses, projections, inclusion maps, spectra, invariant measures and conditional expectations.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1993 .W358. Source: Dissertation Abstracts International, Volume: 54-09, Section: B, page: 4762. Adviser: Chi Song Wong. Thesis (Ph.D.)--University of Windsor (Canada), 1993.