Simultaneous Envelopes for Multivariate Linear Regression

Abstract

We introduce envelopes for simultaneously reducing the predictors and the responses in multivariate linear regression, so the regression then depends only on estimated linear combinations of X and Y. We use a likelihood-based objective function for estimating envelopes and then propose algorithms for estimation of a simultaneous envelope as well as for basic Grassmann manifold optimization. The asymptotic properties of the resulting estimator are studied under normality and extended to general distributions. We also investigate likelihood ratio tests and information criteria for determining the simultaneous envelope dimensions. Simulation studies and real data examples show substantial gain over the classical methods, like partial least squares, canonical correlation analysis, and reduced-rank regression. This article has supplementary material available online.

• Canonical correlations
• Envelope model
• Grassmann manifold
• Partial least squares
• Principal component analysis
• Reduced-rank regression
• Sufficient dimension reduction

ACKNOWLEDGMENTS

We thank the Editor, the Associate Editor, and three referees for their insightful comments and constructive suggestions that have greatly improved the presentation of the article. Research for this article was supported in part by grant DMS-1007547 from the National Science Foundation.