Structured variable selection via prior-induced hierarchical penalty functions

Abstract

The paper studies a grouped variable selection problem in a linear regression setting by proposing a hierarchical penalty function to model collective behavior of the regression coefficients. This hierarchical penalty function consists of two levels. At the top level, it models the group effect of covariates by introducing an index function on the event that the l2-norm of the corresponding regression coefficients is not equal to zero. At the bottom level, it models the individual effect of a covariate with an index function on the event that the corresponding regression coefficient is not equal to zero. Under this hierarchical penalty function, model estimation can be conducted by applying an iteration-based numerical procedure to solve a sequence of modified optimization problems. Simulation study shows that the proposed estimator performs relatively well when the number of covariates exceeds the sample size, and when both the true and false covariates are included in the same group. Theoretical analysis suggests that the l2 estimation error of the proposed estimator can achieve a good upper bound if some regularity conditions are satisfied.