On the adaptive elastic-net with a diverging number of parameters

TL;DR

This paper introduces the adaptive elastic-net, a method designed for high-dimensional model selection that achieves the oracle property and effectively handles collinearity, outperforming existing methods in simulations.

Contribution

It proposes the adaptive elastic-net, combining quadratic regularization with adaptive lasso, and proves its oracle property under weak conditions.

Findings

Adaptive elastic-net achieves the oracle property.

It better handles collinearity than existing methods.

Simulation results show improved finite sample performance.

Abstract

We consider the problem of model selection and estimation in situations where the number of parameters diverges with the sample size. When the dimension is high, an ideal method should have the oracle property [J. Amer. Statist. Assoc. 96 (2001) 1348--1360] and [Ann. Statist. 32 (2004) 928--961] which ensures the optimal large sample performance. Furthermore, the high-dimensionality often induces the collinearity problem, which should be properly handled by the ideal method. Many existing variable selection methods fail to achieve both goals simultaneously. In this paper, we propose the adaptive elastic-net that combines the strengths of the quadratic regularization and the adaptively weighted lasso shrinkage. Under weak regularity conditions, we establish the oracle property of the adaptive elastic-net. We show by simulations that the adaptive elastic-net deals with the collinearity problem better than the other oracle-like methods, thus enjoying much improved finite sample performance.