  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.
