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Yuli Liang
-- doctoral thesis --
Contributions to Estimation and Testing Block Covariance Structures in Multivariate Normal Models
Abstract
This thesis concerns inference problems in balanced random effects models
with a so-called block circular Toeplitz covariance structure. This class of
covariance structures describes the dependency of some specific multivariate
two-level data when both compound symmetry and circular symmetry
appear simultaneously.
We derive two covariance structures under two different invariance restrictions.
The obtained covariance structures reflect both circularity and
exchangeability present in the data. In particular, estimation in the balanced
random effects with block circular covariance matrices is considered.
The spectral properties of such patterned covariance matrices are provided.
Maximum likelihood estimation is performed through the spectral decomposition
of the patterned covariance matrices. Existence of the explicit maximum
likelihood estimators is discussed and sufficient conditions for obtaining
explicit and unique estimators for the variance-covariance components
are derived. Different restricted models are discussed and the corresponding
maximum likelihood estimators are presented.
This thesis also deals with hypothesis testing of block covariance structures,
especially block circular Toeplitz covariance matrices. We consider
both so-called external tests and internal tests. In the external tests, various
hypotheses about testing block covariance structures, as well as mean
structures, are considered, and the internal tests are concerned with testing
specific covariance parameters given the block circular Toeplitz structure.
Likelihood ratio tests are constructed, and the null distributions of the corresponding
test statistics are derived.
Keywords: Block circular symmetry, covariance parameters, explicit maximum
likelihood estimator, likelihood ratio test, restricted model, Toeplitz
matrix
ISBN 978-91-7649-136-2