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.

Keywords:

• Dimension reduction
• Grassmannian
• Partial envelope model
• Reducing subspace

Acknowledgements

The authors thank for the editor, the associate editor and referees for their suggestions that led to the improvement of an early manuscript.

Additional information

Funding

This research was supported by the National Natural Science Foundation of China (Grant Nos. 11471160, 11101114), the National Statistical Science Research Key Program of China (Grant No. 2013LZ45), the Fundamental Research Funds for the Central Universities (Grant No. 30920130111015), the Jiangsu Provincial Basic Research Program (Natural Science Foundation) (Grant No. BK20131345) and the National Statistical Science Research Major Program of China (Grant No. 2018LD01), and sponsored by Qing Lan Project.