Efficient simultaneous partial envelope model in multivariate linear regression

In this paper, we adopt dimension reduction ideas of the partial envelope model proposed by Su and Cook [Partial envelopes for efficient estimation in multivariate linear regression. Biometrika. 2011;98:133- 146] to centre on some predictors of special interest. Our goal is to associate their advantages by simultaneously reducing the predictors of special interest X-1 and the responses Y to decrease both predictive and estimative variation. Motivated by the research results of Cook and Zhang [Simultaneous envelopes for multivariate linear regression. Technometrics. 2015;57:11-25], we propose the simultaneous partial envelope model which can dramatically improve the efficiency of parameter estimation. Further, weshow the maximum likelihood estimators for simultaneous partial envelope model parameters. Meanwhile, we give the asymptotic distribution and theoretical properties, selection of rank and envelopes dimension. At last, the simulation results and real data analysis demonstrate that the performance of the simultaneous partial envelope estimators is much better than that of the other four methods, including ordinary least squares, X-1-envelope, Y-envelope and simultaneous envelope.