UBLUE for the regular two-stage linear model from the perspective of projection operators

The linear models in multi stages and in particular the two-stage models, have great potential that may be used in many scientific research. Nevertheless more than 20 years have elapsed since the two-stage model was defined by [5] and still the use of these models is not yet popular among applied research workers, which we attribute to the complicated expressions of the estimators of the mean vector and other parameters of the model obtained till now in the different papers published on this subject. This article gives alternative expressions for UBLUE of and in the regular two-stage linear model, using projection operators onto and a linear transformation F of the observable random vector y which is linearly sufficient in order to provide new insights and facilitate the use of these models in applied research.