Groupwise partial envelope model: efficient estimation in multivariate linear regression

Abstract

In order to incorporate group information for different groups, we extend the partial envelopes (Su and Cook, Statistica Sinica 23:213–30, 2011) to the groupwise partial envelopes which can improve the efficiency of parameter estimation and enlarge the scope of partial envelope model. It maintains the potential of the original partial envelope methods to increase efficiency and allows for both different regression coefficients and different error structures for diverse groups. Further, we demonstrate the maximum likelihood estimation under the groupwise partial envelope model. Meanwhile, we give asymptotic distribution and theoretical properties. At last, simulation studies are carried out to compare our proposed groupwise partial envelope model with the other three methods, including the standard model, the partial envelope model and the separate partial envelope model. From the simulation results and real data analysis, we can see that the performance of the groupwise partial envelope estimators is much better than that of the standard model estimators, the partial envelope estimators and the separate partial envelope estimators.