Cochran Theorems for Multivariate Components of Variance Models

Let$M_{n\times p}$be the set of all n × p matrix over the real field. For the multivariate normal matrix Y in$M_{n\times p}$with mean μ and covariance$\sum_{j=1}^{k}V_{j}\otimes \Sigma _{j}$, the necessary and sufficient conditions under which$``\{Y^{\prime }W_{i}Y\}_{i=1}^{L}$(with nonnegative defi...

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Bibliographic Details
Published in:Sankhya. Series A 1996-06, Vol.58 (2), p.328-342
Main Authors: Wang, Tonghui, Fu, Ronglin, Wong, Chi Song
Format: Article
Language:English
Subjects:
Covariance
Covariance matrices
Mathematical theorems
Matrices
Maximum likelihood estimation
Real numbers
Statistical variance
Online Access:Get full text
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Summary:Let$M_{n\times p}$be the set of all n × p matrix over the real field. For the multivariate normal matrix Y in$M_{n\times p}$with mean μ and covariance$\sum_{j=1}^{k}V_{j}\otimes \Sigma _{j}$, the necessary and sufficient conditions under which$``\{Y^{\prime }W_{i}Y\}_{i=1}^{L}$(with nonnegative definite$W_{i}$) is a family of independent Wishart$W_{p}(m_{i},\Sigma,\lambda _{i})$random matrices$Y^{\prime }W_{i}Y\text{'}\text{'}$is obtained. The Cochran theorem is extended further to include the case where the set of quadratic forms,$\{Q_{i}(Y)\}$, is a family of the matrix second degree polynomials$Q_{i}(Y)=Y^{\prime }W_{i}Y+B_{i}^{\prime }Y+Y^{\prime }B_{i}+D_{i}$with$B_{i}\in M_{n\times p}$and$D_{i}\in M_{p\times p}$. For illustration, our results are applied to MANOVA models.
ISSN:0581-572X