# Smooth Adjustment for Correlated Effects

## TL;DR

This paper introduces the SACE and GSACE methods, which use a novel adaptive 'reversed' penalty to improve high-dimensional linear regression estimation accuracy amidst complex predictor correlations.

## Contribution

It proposes new regularization methods that reduce bias and false negatives, with less dependence on initial estimators, and proves their oracle properties.

## Key findings

- Methods outperform traditional estimators in simulations.
- Accurately estimate coefficients in highly correlated predictor scenarios.
- Show improved variable selection accuracy.

## Abstract

This paper considers a high dimensional linear regression model with corrected variables. A variety of methods have been developed in recent years, yet it is still challenging to keep accurate estimation when there are complex correlation structures among predictors and the response. We propose an adaptive and "reversed" penalty for regularization to solve this problem. This penalty doesn't shrink variables but focuses on removing the shrinkage bias and encouraging grouping effect. Combining the l_1 penalty and the Minimax Concave Penalty (MCP), we propose two methods called Smooth Adjustment for Correlated Effects (SACE) and Generalized Smooth Adjustment for Correlated Effects (GSACE). Compared with the traditional adaptive estimator, the proposed methods have less influence from the initial estimator and can reduce the false negatives of the initial estimation. The proposed methods can be seen as linear functions of the new penalty's tuning parameter, and are shown to estimate the coefficients accurately in both extremely highly correlated variables situation and weakly correlated variables situation. Under mild regularity conditions we prove that the methods satisfy certain oracle property. We show by simulations and applications that the proposed methods often outperforms other methods.