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Models with a Kronecker Product Covariance Structure: Estimation and Testing
Stockholm University, Faculty of Social Sciences, Department of Statistics.
##### Abstract [en]

In this article we consider a $pq$-dimensional random vector $x$ distributed normally with mean vector $\theta$ and the covariance matrix $\Lambda$, assumed to be positive definite. On the basis of $N$ independent observations on the random vector $x$, we wish to estimate parameters and test the hypothesis $H: \Lambda=\Psi\otimes\Sigma$, where $\Psi = (\psi_{ij}) : q\times q$ and $\Sigma = (\sigma_{ij}) : p\times p$, and $\Lambda = (\psi_{ij}\Sigma)$, the Kronecker product of $\Psi$ and $\Sigma$. That is instead of $\frac{1}{2}pq(pq+1)$ parameters, it has only $\frac{1}{2}p(p + 1) + \frac{1}{2}q(q + 1) - 1$ parameters. When this model holds, we test the hypothesis that $\Psi$ is an identity matrix, a diagonal matrix or of intraclass correlation structure. The maximum likelihood estimators (MLE) are obtained under the hypothesis as well as under the alternatives. Using these estimators the likelihood ratio tests (LRT) are obtained. Moreover, it is shown that the estimators are unique.

2007.
##### Keyword [en]
Covariance structure, flip-flop algorithm, intraclass correlation structure, Kronecker product structure, likelihood ratio test, maximum likelihood estimators, repeated measurements.
##### National Category
Probability Theory and Statistics
##### Identifiers
OAI: oai:DiVA.org:su-13933DiVA: diva2:180453
Available from: 2008-06-24 Created: 2008-06-24Bibliographically approved

#### Open Access in DiVA

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http://biostochastics.slu.se/publikationer/dokument/Report2007_7.pdf

#### Search in DiVA

Nahtman, Tatjana
##### By organisation
Department of Statistics
##### On the subject
Probability Theory and Statistics

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#### Altmetric score

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