Post Selection Shrinkage Estimation for High Dimensional Data Analysis

TL;DR

This paper introduces a post selection shrinkage estimator (PSE) for high-dimensional data that enhances prediction accuracy by adaptively shrinking a ridge estimator after variable selection, outperforming existing methods.

Contribution

It proposes a novel post selection shrinkage estimation method that improves prediction performance in high-dimensional settings, especially when covariates have weak effects.

Findings

PSE outperforms weighted ridge estimators in prediction accuracy.

PSE significantly improves predictions over existing Lasso-type methods.

Simulation and real data analyses confirm the effectiveness of PSE.

Abstract

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.