Envelope-based sparse reduced-rank regression for multivariate linear model

Envelope models were first proposed by Cook et al. (2010) as a method to reduce estimative and predictive variations in multivariate regression. Sparse reduced-rank regression, introduced by Chen and Huang (2012), is a widely used technique that performs dimension reduction and variable selection simultaneously in multivariate regression. In this work, we combine envelope models and sparse reduced-rank regression method to propose an envelope-based sparse reduced-rank regression estimator, and then establish its consistency, asymptotic normality and oracle property in high-dimensional data. We carry out some Monte Carlo simulation studies and also analyze two datasets to demonstrate that the proposed envelope-based sparse reduced-rank regression method displays good variable selection and prediction performance.