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Block Circular Symmetry in Multilevel Models
Stockholm University, Faculty of Social Sciences, Department of Statistics.
Stockholm University, Faculty of Social Sciences, Department of Statistics.
2011 (English)Report (Other academic)
Abstract [en]

Models that describe symmetries present in the error structure of observations have been widely used in dierent applications, with early examples from psychometric and medical research. The aim of this article is to study a multilevel model with a covariance structure that is block circular symmetric. Useful results are obtained for the spectra of these structured matrices.

Place, publisher, year, edition, pages
Stockholm: Department of Statistics, Stockholm University , 2011. , 23 p.
Series
Research Report / Department of Statistics, Stockholm University, ISSN 0280-7564 ; 3
Keyword [en]
Covariance matrix, Circular block symmetry, Multilevel model, Symmetry model, Spectrum
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
URN: urn:nbn:se:su:diva-72075OAI: oai:DiVA.org:su-72075DiVA: diva2:488906
Available from: 2012-09-10 Created: 2012-02-02 Last updated: 2015-03-25Bibliographically approved
In thesis
1. A study of multilevel models with block circular  symmetric covariance structures
Open this publication in new window or tab >>A study of multilevel models with block circular  symmetric covariance structures
2011 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns the study of multilevel models with specific patterned covariance structures and addresses the issues of maximum likelihoodestimation. In particular, circular symmetric hierarchical datastructures are considered.

In the first paper (Paper I), we consider so-called dihedral block symmetrymodels and extend them to models which covariance structures reflects both circularity and exchangeability present in the data. The main contribution of Paper I are two derived patterns of the covariancematrices which characterizes models under consideration. The relationship between these two patterned covariance matrices was investigatedand it has been verified they are similar matrices. New expressions for the eigenvalues of block circular symmetric matrices are obtained which take into account the block structure. Paper II deals with estimation ofbalanced multilevel models with block circular symmetric covariance matrices. The spectral properties of such patterned covariance matrices are established. Maximum likelihood estimation is considered through the spectral decomposition of the patterned covariance matrix. The main results of Paper II concern the spectrum of the covariance matrix inthe model of interest and the existence of explicit maximum likelihood estimators for the covariance parameters.

Place, publisher, year, edition, pages
Stockholm: Department of Statistics, Stockholm University, 2011. 22 p.
Keyword
Block circular symmetry, Covariance matrix, Explicit solution, Maximum likelihood estimator, Multilevel model, Spectrum
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:su:diva-81538 (URN)978-91-633-9851-3 (ISBN)
Opponent
Supervisors
Available from: 2013-01-25 Created: 2012-10-25 Last updated: 2013-01-25Bibliographically approved
2. Contributions to Estimation and Testing Block Covariance Structures in Multivariate Normal Models
Open this publication in new window or tab >>Contributions to Estimation and Testing Block Covariance Structures in Multivariate Normal Models
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Stockholm: Department of Statistics, Stockholm Univeristy, 2015. 54 p.
Keyword
Block circular symmetry, covariance parameters, explicit maximum likelihood estimator, likelihood ratio test, restricted model, Toeplitz matrix
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:su:diva-115347 (URN)978-91-7649-136-2 (ISBN)
Public defence
2015-05-11, De Geersalen, Geovetenskapens hus, Svante Arrhenius väg 14, Stockholm, 10:00 (English)
Opponent
Supervisors
Available from: 2015-04-17 Created: 2015-03-19 Last updated: 2015-04-20Bibliographically approved

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