Equivalent Linear Models to Reduce Computations

Abstract

Any linear model can be expressed as many different linear models, all of which yield identical first and second moments of the data vector. The models of such a set are defined as linearly equivalent. As a consequence, best linear unbiased estimators and best linear unbiased predictors derived from one model can be converted by simple linear transformations to best linear unbiased estimators and best linear unbiased predictors of the elements of any equivalent model. Also, translation invariant, quadratic, unbiased estimators of variances, and covariances can be converted to comparable estimates of the variances and covariances of any equivalent model. Consequently, if computations are too burdensome for a particular data set and a particular model, it may be possible to find an equivalent model that is computationally feasible. Four examples of sets of equivalent models are described and simple numerical examples are used to illustrate best linear unbiased estimators and best linear unbiased predictors for these models.