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Estimation in Multivariate Linear Models with Linearly Structured Covariance Matrices
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
2012 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis focuses on the problem of estimating parameters in multivariate linear models where particularly the mean has a bilinear structure and the covariance matrix has a linear structure. Most of techniques in statistical modeling rely on the assumption that data were generated from the normal distribution. Whereas real data may not be exactly normal, the normal distributions serve as a useful approximation to the true distribution. The modeling of normally distributed data relies heavily on the estimation of the mean and the covariance matrix. The interest of considering various structures for the covariance matrices in different statistical models is partly driven by the idea that altering the covariance structure of a parametric model alters the variances of the model’s estimated mean parameters.

The extended growth curve model with two terms and a linearly structured covariance matrix is considered. In general there is no problem to estimate the covariance matrix when it is completely unknown. However, problems arise when one has to take into account that there exists a structure generated by a few number of parameters. An estimation procedure that handles linear structured covariance matrices is proposed. The idea is first to estimate the covariance matrix when it should be used to define an inner product in a regression space and thereafter reestimate it when it should be interpreted as a dispersion matrix. This idea is exploited by decomposing the residual space, the orthogonal complement to the design space, into three orthogonal subspaces. Studying residuals obtained from projections of observations on these subspaces yields explicit consistent estimators of the covariance matrix. An explicit consistent estimator of the mean is also proposed and numerical examples are given.

The models based on normally distributed random matrix are also studied in this thesis. For these models, the dispersion matrix has the so called Kronecker product structure and they can be used for example to model data with spatio-temporal relationships. The aim is to estimate the parameters of the model when, in addition, one covariance matrix is assumed to be linearly structured. On the basis of n independent observations from a matrix normal distribution, estimation equations in a flip-flop relation are presented and numerical examples are given.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2012. , 25 p.
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1531
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-78845Local ID: LIU-TEK-LIC-2012:16ISBN: 978-91-7519-886-6 (print)OAI: oai:DiVA.org:liu-78845DiVA: diva2:536195
Presentation
2012-06-08, BL32 (Nobel), hus B, Campus Valla, Linköpings universitet, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2012-06-21 Created: 2012-06-21 Last updated: 2014-09-29Bibliographically approved
List of papers
1. Estimation of parameters in the extended growth curve model with a linearly structured covariance matrix
Open this publication in new window or tab >>Estimation of parameters in the extended growth curve model with a linearly structured covariance matrix
2012 (English)In: Acta et Commentationes Universitatis Tartuensis de Mathematica, ISSN 1406-2283, E-ISSN 2228-4699, Vol. 16, no 1, 13-32 p.Article in journal (Refereed) Published
Abstract [en]

In this paper the extended growth curve model with two terms and a linearly structured covariance matrix is considered. We propose an estimation procedure that handles linear structured covariance matrices. The idea is first to estimate the covariance matrix when it should be used to define an inner product in a regression space and thereafter reestimate it when it should be interpreted as a dispersion matrix. This idea is exploited by decomposing the residual space, the orthogonal complement to the design space, into three orthogonal subspaces. Studying residuals obtained from projections of observations on these subspaces yields explicit consistent estimators of the covariance matrix. An explicit consistent estimator of the mean is also proposed and numerical examples are given.

Keyword
Extended growth curve model, estimation, linearly structured covariance matrix, residuals
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-73218 (URN)
Available from: 2011-12-22 Created: 2011-12-22 Last updated: 2017-12-08Bibliographically approved
2. Estimation in multivariate linear models with Kronecker product and linear structures on the covariance matrices
Open this publication in new window or tab >>Estimation in multivariate linear models with Kronecker product and linear structures on the covariance matrices
(English)Manuscript (preprint) (Other academic)
Abstract [en]

This paper deals with models based on normally distributed random matrices. More specifically the model considered is X ∼ Np,q(M, Σ, Ψ) with mean M, a p×q matrix, assumed to follow a bilinear structure, i.e., E[X] = M = ABC, where A and C are known design matrices, B is unkown parameter matrix, and the dispersion matrix of X has a Kronecker product structure, i.e., D[X] = Ψ ⊗ Σ, where both Ψ and Σ are unknown positive definite matrices. The model may be used for example to model data with spatiotemporal relationships. The aim is to estimate the parameters of the model when, in addition, Σ is assumed to be linearly structured. In the paper, on the basis of n independent observations on the random matrix X, estimation equations in a flip-flop relation are presented and numerical examples are given.

Keyword
Growth curve model, maximum likelihood estimators, estimation equations, flip-flop algorithm, Kronecker product structure, linear structure
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-78844 (URN)LiTH-MAT-R-2012/05-SE (ISRN)
Available from: 2012-06-21 Created: 2012-06-21 Last updated: 2014-09-29Bibliographically approved

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Nzabanita, Joseph

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