A modified partial envelope tensor response regression

The envelope model is a useful statistical technique that can be applied to multivariate linear regression problems. It aims to remove immaterial information via sufficient dimension reduction techniques while still gaining efficiency and providing accurate parameter estimates. Recently, envelope tensor versions have been developed to extend this technique to tensor data. In this work, a partial tensor envelope model is proposed that allows for a parsimonious version of tensor response regression when only certain predictors are of interest. The consistency and asymptotic normality of the regression coefficients estimator are also established theoretically, which provides a rigorous foundation for the proposed method. In numerical studies using both simulated and real‐world data, the partial tensor envelope model is shown to outperform several existing methods in terms of the efficiency of the regression coefficients associated with the selected predictors.