Sankhya: The Indian Journal of Statistics

2005, Volume 67, Pt. 1, 74--89

Some Further Results Concerning the Decomposition of the Watson Efficiency in Partitioned Linear Models


K.L. Chu, McGill University, Montr\'eal, Qu\'ebec, Canada
J. Isotalo, S. Puntanen, University of Tampere, Tampere, Finland
G.P.H. Styan, McGill University, Montr\'eal, Qu\'ebec, Canada

SUMMARY. While considering  the estimation of regression coefficients in a partitioned weakly singular linear model, Chu, Isotalo, Puntanen and Styan (2004a) introduced a particular decomposition for the  Watson efficiency of the ordinary least squares estimator. This decomposition presents the ``total'' Watson efficiency as a product of three factors. In this paper we give new insight into the decomposition showing that all three factors are related to the efficiencies of particular submodels or their transformed versions. Moreover, we prove an interesting connection between a particular reduction of the Watson efficiency and the concept of linear sufficiency. We shortly review the relation between the efficiency and specific canonical correlations. We also introduce the corresponding decomposition for the Bloom\-field--Watson commutator criterion, and give a necessary and sufficient condition for its specific  eduction.

AMS (1991) subject classification. 62J05, 62H12, 62H20.

Key words and phrases. Best linear unbiased estimation, Watson efficiency, Gau\ss--Markov model, linear sufficiency, partitioned linear model, reduced linear model.

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