High-dimensional Bayesian Tobit regression for censored response with Horseshoe prior

TL;DR

This paper introduces a Bayesian high-dimensional Tobit regression model using Horseshoe priors, providing theoretical guarantees and demonstrating superior performance over existing methods in censored data analysis.

Contribution

It develops the first Bayesian high-dimensional Tobit model with theoretical consistency results and efficient Gibbs sampling, addressing sparsity and censoring simultaneously.

Findings

Outperforms recent Lasso-Tobit methods in simulations

Establishes posterior consistency and concentration rates

Provides an R package for implementation

Abstract

Censored response variables--where outcomes are only partially observed due to known bounds--arise in numerous scientific domains and present serious challenges for regression analysis. The Tobit model, a classical solution for handling left-censoring, has been widely used in economics and beyond. However, with the increasing prevalence of high-dimensional data, where the number of covariates exceeds the sample size, traditional Tobit methods become inadequate. While frequentist approaches for high-dimensional Tobit regression have recently been developed, notably through Lasso-based estimators, the Bayesian literature remains sparse and lacks theoretical guarantees. In this work, we propose a novel Bayesian framework for high-dimensional Tobit regression that addresses both censoring and sparsity. Our method leverages the Horseshoe prior to induce shrinkage and employs a data augmentation strategy to facilitate efficient posterior computation via Gibbs sampling. We establish posterior consistency and derive concentration rates under sparsity, providing the first theoretical results for Bayesian Tobit models in high dimensions. Numerical experiments show that our approach outperforms favorably with the recent Lasso-Tobit method. Our method is implemented in the R package tobitbayes, which can be found on Github.