Foundations for Envelope Models and Methods

Abstract

Envelopes were recently proposed by Cook, Li and Chiaromonte as a method for reducing estimative and predictive variations in multivariate linear regression. We extend their formulation, proposing a general definition of an envelope and a general framework for adapting envelope methods to any estimation procedure. We apply the new envelope methods to weighted least squares, generalized linear models and Cox regression. Simulations and illustrative data analysis show the potential for envelope methods to significantly improve standard methods in linear discriminant analysis, logistic regression and Poisson regression. Supplementary materials for this article are available online.

• Generalized linear models
• Grassmannians
• Weighted least squares

Additional information

Notes on contributors

R. Dennis Cook

R. Dennis Cook is Professor, School of Statistics, University of Minnesota, Minneapolis, MN 55455 (E-mail: [email protected]).

Xin Zhang

Xin Zhang is Assistant Professor, Department of Statistics, Florida State University, Tallahassee, FL 32306 (E-mail: [email protected]). Research for this article was supported in part by grant DMS-1007547 from the National Science Foundation. The authors thank Doug Hawkins for providing the Colon Cancer Diagnosis dataset. The authors are grateful to the Editor, an Associate Editor, and a referee for comments that led to significant improvements in this article and to Lexin Li for helpful discussions.