High-Dimensional Censored Regression via the Penalized Tobit Likelihood

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 majorization 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 ${\mathcal{l}}_2$ 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 on high-dimensional left-censored data. 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. A supplementary file contains intermediate theoretical results and technical proofs.

Keywords:

• Censored regression
• Coordinate descent
• Folded concave penalty
• High dimensions
• Strong oracle property
• Tobit model

Supplementary Materials

penalized_tobit_proofs:

This supplementary file contains intermediate results and technical proofs for the results in this article.

Acknowledgments

We thank the referees and the editor whose thoughtful comments significantly improved this article.

Additional information

Funding

This work is supported in part by NSF DMS 1915842 and 2015120.