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On the Distribution of Matrix Quadratic Forms
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.ORCID iD: 0000-0001-9896-4438
Royal Institute of Technology, Sweden.
2012 (English)In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 41, no 18, 3403-315 p.Article in journal (Refereed) Published
Abstract [en]

A characterization of the distribution of the multivariate quadratic form given by XAX′, where X is a p×n normally distributed matrix and A is an n×n symmetric real matrix, is presented. We show that the distribution of the quadratic form is the same as the distribution of a weighted sum of noncentralWishart distributed matrices. This is applied to derive the distribution of the sample covariance between the rows of X when the expectation is the same for every column and is estimated with the regular mean.

Place, publisher, year, edition, pages
Taylor & Francis, 2012. Vol. 41, no 18, 3403-315 p.
Keyword [en]
Quadratic form; Spectral decomposition; Eigenvalues; Singular matrix normal distribution; Non-centralWishart distribution
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-18513DOI: 10.1080/03610926.2011.563009ISI: 000308465400007OAI: oai:DiVA.org:liu-18513DiVA: diva2:220119
Available from: 2009-05-29 Created: 2009-05-29 Last updated: 2017-12-13
In thesis
1. Studies in Estimation of Patterned Covariance Matrices
Open this publication in new window or tab >>Studies in Estimation of Patterned Covariance Matrices
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Many testing, estimation and confidence interval procedures discussed in the multivariate statistical literature are based on the assumption that the observation vectors are independent and normally distributed. The main reason for this is that often sets of multivariate observations are, at least approximately, normally distributed. Normally distributed data can be modeled entirely in terms of their means and variances/covariances. Estimating the mean and the covariance matrix is therefore a problem of great interest in statistics and it is of great significance to consider the correct statistical model. The estimator for the covariance matrix is important since inference on the mean parameters strongly depends on the estimated covariance matrix and the dispersion matrix for the estimator of the mean is a function of it.

In this thesis the problem of estimating parameters for a matrix normal distribution with different patterned covariance matrices, i.e., different statistical models, is studied.

A p-dimensional random vector is considered for a banded covariance structure reflecting m-dependence. A simple non-iterative estimation procedure is suggested which gives an explicit, unbiased and consistent estimator of the mean and an explicit and consistent estimator of the covariance matrix for arbitrary p and m.

Estimation of parameters in the classical Growth Curve model when the covariance matrix has some specific linear structure is considered. In our examples maximum likelihood estimators can not be obtained explicitly and must rely on numerical optimization algorithms. Therefore explicit estimators are obtained as alternatives to the maximum likelihood estimators. From a discussion about residuals, a simple non-iterative estimation procedure is suggested which gives explicit and consistent estimators of both the mean and the linearly structured covariance matrix.

This thesis also deals with the problem of estimating the Kronecker product structure. The sample observation matrix is assumed to follow a matrix normal distribution with a separable covariance matrix, in other words it can be written as a Kronecker product of two positive definite matrices. The proposed estimators are used to derive a likelihood ratio test for spatial independence. Two cases are considered, when the temporal covariance is known and when it is unknown. When the temporal covariance is known, the maximum likelihood estimates are computed and the asymptotic null distribution is given. In the case when the temporal covariance is unknown the maximum likelihood estimates of the parameters are found by an iterative alternating algorithm and the null distribution for the likelihood ratio statistic is discussed.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2009. 56 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1255
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-18519 (URN)978-91-7393-622-4 (ISBN)
Public defence
2009-05-29, Nobel (BL32), B-huset, ingång 23, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2009-05-29 Created: 2009-05-29 Last updated: 2014-09-29Bibliographically approved

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Ohlson, MartinKoski, Timo

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