Regularization Methods Based on the $L_q$-Likelihood for Linear Models with Heavy-Tailed Errors

TL;DR

This paper introduces regularization techniques for linear models with heavy-tailed errors based on the $L_q$-likelihood, unifying with $q$-normal distributions, and demonstrates their effectiveness through numerical experiments.

Contribution

It proposes a novel regularization framework using $L_q$-likelihood for heavy-tailed error models, aligning with existing methods for $q$-normal distributions.

Findings

Methods perform well with heavy-tailed errors

Efficient computation using existing packages

Consistent with regularization for normal errors

Abstract

We propose regularization methods for linear models based on the $L_q$-likelihood, which is a generalization of the log-likelihood using a power function. Some heavy-tailed distributions are known as $q$-normal distributions. We find that the proposed methods for linear models with $q$-normal errors coincide with the regularization methods that are applied to the normal linear model. The proposed methods work well and efficiently, and can be computed using existing packages. We examine the proposed methods using numerical experiments, showing that the methods perform well, even when the error is heavy-tailed.