Lasso Penalization for High-Dimensional Beta Regression Models: Computation, Analysis, and Inference

TL;DR

This paper develops a theoretical framework for LASSO penalization in high-dimensional beta regression, addressing non-convexity issues, providing error bounds, and proposing a debiasing method for confidence intervals, validated through simulations and real data.

Contribution

It introduces a novel analysis of LASSO in high-dimensional beta regression, including error bounds and a debiasing approach for inference, which were previously lacking.

Findings

Derived non-asymptotic $ ext{l}_1$-error bounds for stationary points.

Proposed a debiasing method for confidence interval construction.

Validated the methodology with simulations and real data application.

Abstract

Beta regression is commonly employed when the outcome variable is a proportion. Since its conception, the approach has been widely used in applications spanning various scientific fields. A series of extensions have been proposed over time, several of which address variable selection and penalized estimation, e.g., with an $\ell_1$-penalty (LASSO). However, a theoretical analysis of this popular approach in the context of Beta regression with high-dimensional predictors is lacking. In this paper, we aim to close this gap. A particular challenge arises from the non-convexity of the associated negative log-likelihood, which we address by resorting to a framework for analyzing stationary points in a neighborhood of the target parameter. Leveraging this framework, we derive a non-asymptotic bound on the $\ell_1$-error of such stationary points. In addition, we propose a debiasing approach to construct confidence intervals for the regression parameters. A proximal gradient algorithm is devised for optimizing the resulting penalized negative log-likelihood function. Our theoretical analysis is corroborated via simulation studies, and a real data example concerning the prediction of county-level proportions of incarceration is presented to showcase the practical utility of our methodology.