The adaptive Gril estimator with a diverging number of parameters

TL;DR

This paper introduces the AdaGril estimator, an adaptive regularization method for high-dimensional linear regression that effectively handles collinearity and variable selection, with proven oracle properties and superior simulation performance.

Contribution

It extends the adaptive Elastic Net to the AdaGril, providing a new estimator with oracle properties and improved variable selection in diverging parameter scenarios.

Findings

AdaGril achieves oracle property under weak conditions.

It effectively handles collinearity in high-dimensional models.

Simulation results show AdaGril outperforms competitors.

Abstract

We consider the problem of variables selection and estimation in linear regression model in situations where the number of parameters diverges with the sample size. We propose the adaptive Generalized Ridge-Lasso (\mbox{AdaGril}) which is an extension of the the adaptive Elastic Net. AdaGril incorporates information redundancy among correlated variables for model selection and estimation. It combines the strengths of the quadratic regularization and the adaptively weighted Lasso shrinkage. In this paper, we highlight the grouped selection property for AdaCnet method (one type of AdaGril) in the equal correlation case. Under weak conditions, we establish the oracle property of AdaGril which ensures the optimal large performance when the dimension is high. Consequently, it achieves both goals of handling the problem of collinearity in high dimension and enjoys the oracle property. Moreover, we show that AdaGril estimator achieves a Sparsity Inequality, i. e., a bound in terms of the number of non-zero components of the 'true' regression coefficient. This bound is obtained under a similar weak Restricted Eigenvalue (RE) condition used for Lasso. Simulations studies show that some particular cases of AdaGril outperform its competitors.