On a singular partitioned linear model and some associated reduced models

摘要

In this paper we consider a partitioned linear model (Y, X_1β_1 + X_2β_2, σ~2V) where X = (X_1:X_2) and V may be rank deficient. We also consider some associated reduced models involving only β_2 (and not β_1) and identify when each of these models is valid. We extend a result of Bhimasankaram and Saha Ray (1995) to the singular case. THe case of equality constraints on β_2 is considered and disposed easily since it can be transformed to the case with no constraints (with a singular dispersion matrix). A residual interpretation is given to a result proved in Bhimasankaram and Sengupta (1995) and to the result proved in this paper which can be seen as an extension of standard practice in the usual multiple linear regression.