liu.seSearch for publications in DiVA
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Estimation of parameters in the extended growth curve model with a linearly structured covariance matrix
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.ORCID iD: 0000-0001-9896-4438
Swedish University of Agricultural Sciences.
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.

Place, publisher, year, edition, pages
2012. Vol. 16, no 1, 13-32 p.
Keyword [en]
Extended growth curve model, estimation, linearly structured covariance matrix, residuals
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:liu:diva-73218OAI: oai:DiVA.org:liu-73218DiVA: diva2:469199
Available from: 2011-12-22 Created: 2011-12-22 Last updated: 2017-12-08Bibliographically approved
In thesis
1. Estimation in Multivariate Linear Models with Linearly Structured Covariance Matrices
Open this publication in new window or tab >>Estimation in Multivariate Linear Models with Linearly Structured Covariance Matrices
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:nbn:se:liu:diva-78845 (URN)LIU-TEK-LIC-2012:16 (Local ID)978-91-7519-886-6 (ISBN)LIU-TEK-LIC-2012:16 (Archive number)LIU-TEK-LIC-2012:16 (OAI)
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
2. Bilinear and Trilinear Regression Models with Structured Covariance Matrices
Open this publication in new window or tab >>Bilinear and Trilinear Regression Models with Structured Covariance Matrices
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis focuses on the problem of estimating parameters in bilinear and trilinear regression models in which random errors are normally distributed. In these models the covariance matrix has a Kronecker product structure and some factor matrices may be linearly structured. 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.

Firstly, the extended growth curve model with a linearly structured covariance matrix is considered. The main theme is to find explicit estimators for the mean and for the linearly structured covariance matrix. We show how to decompose the residual space, the orthogonal complement to the mean space, into appropriate orthogonal subspaces and how to derive explicit estimators of the covariance matrix from the sum of squared residuals obtained by projecting observations on those subspaces. Also an explicit estimator of the mean is derived and some properties of the proposed estimators are studied.

Secondly, we study a bilinear regression model with matrix normally distributed random errors. For those models, the dispersion matrix follows a Kronecker product structure and it 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, estimating equations, a flip-flop relation, are established.

At last, the models based on normally distributed random third order tensors are studied. These models are useful in analyzing 3-dimensional data arrays. In some studies the analysis is done using the tensor normal model, where the focus is on the estimation of the variance-covariance matrix which has a Kronecker structure. Little attention is paid to the structure of the mean, however, there is a potential to improve the analysis by assuming a structured mean. We formally introduce a 2-fold growth curve model by assuming a trilinear structure for the mean in the tensor normal model and propose an estimation algorithm for parameters. Also some extensions are discussed.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2015. 36 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1665
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-118089 (URN)10.3384/diss.diva-118089 (DOI)978-91-7519-070-9 (ISBN)
Public defence
2015-06-11, BL32, B-huset, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2015-05-21 Created: 2015-05-21 Last updated: 2015-05-21Bibliographically approved

Open Access in DiVA

No full text

Other links

Link to article in full text

Authority records BETA

Nzabanita, JosephSingull, Martinvon Rosen, Dietrich

Search in DiVA

By author/editor
Nzabanita, JosephSingull, Martinvon Rosen, Dietrich
By organisation
Mathematical Statistics The Institute of Technology
In the same journal
Acta et Commentationes Universitatis Tartuensis de Mathematica
Probability Theory and Statistics

Search outside of DiVA

GoogleGoogle Scholar

urn-nbn

Altmetric score

urn-nbn
Total: 449 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf