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Contributions to Estimation and Testing Block Covariance Structures in Multivariate Normal Models
Stockholm University, Faculty of Social Sciences, Department of Statistics.
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 [en]
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: urn:nbn:se:su:diva-115347ISBN: 978-91-7649-136-2 (print)OAI: oai:DiVA.org:su-115347DiVA: diva2:796644
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
List of papers
1. Block Circular Symmetry in Multilevel Models
Open this publication in new window or tab >>Block Circular Symmetry in Multilevel Models
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
Covariance matrix, Circular block symmetry, Multilevel model, Symmetry model, Spectrum
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:su:diva-72075 (URN)
Available from: 2012-09-10 Created: 2012-02-02 Last updated: 2015-03-25Bibliographically approved
2. On estimation in multilevel models with block circular symmetric covariance structure
Open this publication in new window or tab >>On estimation in multilevel models with block circular symmetric covariance structure
2012 (English)In: Acta et Commentationes Universitatis Tartuensis de Mathematica, ISSN 1406-2283, E-ISSN 2228-4699, Vol. 16, no 1, 83-96 p.Article in journal (Refereed) Published
Abstract [en]

In this article we consider a multilevel model with block circular symmetric covariance structure. Maximum likelihood estimation of the parameters of this model is discussed. We show that explicit maximum likelihood estimators of variance components exist under certain restrictions on the parameter space.

Keyword
Circular block symmetry, constrained model, patterned covariance matrix, explicit maximum likelihood estimator, multilevel model
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:su:diva-84275 (URN)
Available from: 2012-12-20 Created: 2012-12-20 Last updated: 2017-12-06Bibliographically approved
3. On estimation in hierarchical models with block circular covariance structures
Open this publication in new window or tab >>On estimation in hierarchical models with block circular covariance structures
2015 (English)In: Annals of the Institute of Statistical Mathematics, ISSN 0020-3157, E-ISSN 1572-9052, Vol. 67, no 4, 773-791 p.Article in journal (Refereed) Published
Abstract [en]

Hierarchical linear models with a block circular covariance structure are considered. Sufficient conditions for obtaining explicit and unique estimators for the variance–covariance components are derived. Different restricted models are discussed and maximum likelihood estimators are presented. The theory is illustrated through covariance matrices of small sizes and a real-life example.

Keyword
Circular block symmetry, Estimation, Identifiability, Maximum likelihood estimator, Restricted model, Variance components
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:su:diva-107708 (URN)10.1007/s10463-014-0475-8 (DOI)000356541300007 ()
Available from: 2014-09-24 Created: 2014-09-24 Last updated: 2017-12-05Bibliographically approved
4. Testing in multivariate normal models with block circular covariance structures
Open this publication in new window or tab >>Testing in multivariate normal models with block circular covariance structures
2015 (English)Report (Other academic)
Abstract [en]

In this article, the results concerning hypothesis testing in multivariate normal models with block circular covariance structures are obtained. Hypotheses about a general block structure of the covariance matrix and specific covariance parameters have been of main interest. In addition, the tests about patterned mean vectors have been considered.The corresponding likelihood ratio statistics are derived and their null distributions are studied.

Place, publisher, year, edition, pages
Stockholm: Department of Statistics, Stockholm University, 2015
Series
Research Report / Department of Statistics, Stockholm University, ISSN 0280-7564 ; 2
Keyword
Beta random variables, Canonical reduction, Covariance parameters, Likelihood ratio test, Restricted model
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:su:diva-115320 (URN)
Available from: 2015-03-25 Created: 2015-03-19 Last updated: 2015-04-20Bibliographically approved

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