High-dimensional Censored Regression via the Penalized Tobit Likelihood

TL;DR

This paper introduces penalized Tobit models for high-dimensional left-censored regression, providing efficient algorithms, theoretical guarantees, and demonstrating superior predictive performance through simulations and real HIV data analysis.

Contribution

It develops the first penalized Tobit models for high-dimensional censored regression, with algorithms and theoretical analysis ensuring accurate estimation and variable selection.

Findings

Penalized Tobit models outperform existing methods in simulations.

Theoretical bounds established for estimation loss.

Application to HIV data identifies potential drug resistance mutations.

Abstract

High-dimensional regression and regression with a left-censored response are each well-studied topics. In spite of this, few methods have been proposed which deal with both of these complications simultaneously. The Tobit model -- long the standard method for censored regression in economics -- has not been adapted for high-dimensional regression at all. To fill this gap and bring up-to-date techniques from high-dimensional statistics to the field of high-dimensional left-censored regression, we propose several penalized Tobit models. We develop a fast algorithm which combines quadratic minimization with coordinate descent to compute the penalized Tobit solution path. Theoretically, we analyze the Tobit lasso and Tobit with a folded concave penalty, bounding the $\ell_2$ estimation loss for the former and proving that a local linear approximation estimator for the latter possesses the strong oracle property. Through an extensive simulation study, we find that our penalized Tobit models provide more accurate predictions and parameter estimates than other methods. We use a penalized Tobit model to analyze high-dimensional left-censored HIV viral load data from the AIDS Clinical Trials Group and identify potential drug resistance mutations in the HIV genome. Appendices contain intermediate theoretical results and technical proofs.