Design-Assisted Regression

Abstract

We consider regression problems in which the marginal distribution of the covariates is informative for estimation and variable selection, rather than merely auxiliary. Motivated by random-design, high-dimensional, and latent-effect settings, we propose a general design-assisted regression framework in which the estimating criterion depends on both the conditional model for $Y \mid \bfX$ and structured features of the covariate distribution. The framework identifies two roles of design information: stabilizing weak design directions through quadratic regularization and correcting latent-effect distortion through nuisance augmentation. We establish oracle properties for the resulting estimator, separate the effects of stochastic error, shrinkage, and approximation, and compare it with a benchmark sparse procedure that ignores design information. These results show that the proposed framework improves estimation while preserving first-order prediction performance. Numerical studies and two real-data applications illustrate the practical impact of incorporating design information.