  In high-dimensional data settings where $p\gg n$, many penalized
regularization approaches were studied for simultaneous variable selection and
estimation. However, with the existence of covariates with weak effect, many
existing variable selection methods, including Lasso and its generations,
cannot distinguish covariates with weak and no contribution. Thus, prediction
based on a subset model of selected covariates only can be inefficient. In this
paper, we propose a post selection shrinkage estimation strategy to improve the
prediction performance of a selected subset model. Such a post selection
shrinkage estimator (PSE) is data-adaptive and constructed by shrinking a post
selection weighted ridge estimator in the direction of a selected candidate
subset. Under an asymptotic distributional quadratic risk criterion, its
prediction performance is explored analytically. We show that the proposed post
selection PSE performs better than the post selection weighted ridge estimator.
More importantly, it improves the prediction performance of any candidate
subset model selected from most existing Lasso-type variable selection methods
significantly. The relative performance of the post selection PSE is
demonstrated by both simulation studies and real data analysis.

Methodology