Smooth predictive model fitting in regression

We propose a smooth hypothesis-testing type method for model fitting in regression and develop its theoretical properties in a moderately high-dimensional setting. We derive the asymptotic behavior of the L2L2 prediction risk and the optimal choice of the threshold-determining parameter under the Gaussian regression framework with orthogonal covariates. When the covariates are not orthogonal, this method can be used in conjunction with some reasonable procedures for orthogonalizing the covariates, such as a stepwise regression coupled with the Gram–Schmidt procedure. Simulation results show that our proposed method often outperforms its competitors. Asymptotic approximation of the mean square error of the proposed estimator is obtained and it is shown that the optimal rate of convergence depends on the behavior of the smoothing function at zero.