On additive and block decompositions of WLSEs under a multiple partitioned regression model

Abstract

While considering the mechanism of weighted least-squares estimators (WLSEs) of regression coefficients in a partitioned linear model, Tian and Takane [On sum decompositions of weighted least-squares estimators under the partitioned linear model, Comm. Statist. Theory Methods 37 (2008), pp. 55–69] gave some identifying conditions for the WLSEs to be the sum of WLSEs under its two small models based on orthogonality of regressors with respect to the given weight matrix. The purpose of this paper is to show how to establish additive and block decompositions of WLSEs under a multiple partitioned linear model and its k small models based on orthogonality of regressors with respect to a given weight matrix.

Keywords:

• multiple partitioned linear model
• small models
• mean vector
• weighted least-squares estimator
• best linear unbiased estimator
• projector
• additive decomposition
• block decomposition
• V-orthogonality
• matrix rank method

Mathematics Subject Classifications (2000) :

• 62J05
• 62H12
• 15A09

Acknowledgements

The authors are grateful to anonymous referees for their helpful comments and suggestions on an earlier version of this paper.