Linear regression under log-concave and Gaussian scale mixture errors: comparative study

Abstract

Gaussian error distributions are a common choice in traditional regression models for the maximum likelihood (ML) method. However, this distributional assumption is often suspicious especially when the error distribution is skewed or has heavy tails. In both cases, the ML method under normality could break down or lose efficiency. In this paper, we consider the log-concave and Gaussian scale mixture distributions for error distributions. For the log-concave errors, we propose to use a smoothed maximum likelihood estimator for stable and faster computation. Based on this, we perform comparative simulation studies to see the performance of coefficient estimates under normal, Gaussian scale mixture, and log-concave errors. In addition, we also consider real data analysis using Stack loss plant data and Korean labor and income panel data.

Penalized maximum likelihood estimation with symmetric log-concave errors and LASSO penalty

“…(Hu et al, 2017) applied the finite mixture of regressions with each component having a log-concave error density and gained robustness by adopting the idea of least trimmed squares. (Kim and Seo, 2018) compared the performance of estimators when the error distribution is assumed as Gaussian scale mixture and log-concave densities based on numerical studies. (Kim and Seo, 2021) proposed a modal linear regression assuming that the error distribution is logconcave.…”

“…In Section 2, we define the notation and introduce the penalized linear regression models with symmetric log-concave errors. In Section 3, we propose using a smoothed log-concave maximum likelihood estimator (Chen and Samworth, 2013;Kim and Seo, 2018) with a Gaussian kernel for the estimation of the initial regression coefficient. Numerical simulations and real data studies are also conducted to compare the performance of the proposed method with other existing methods in Section 4.…”