  This paper considers inference in a linear regression model with random right
censoring and outliers. The number of outliers can grow with the sample size
while their proportion goes to zero. The model is semiparametric and we make
only very mild assumptions on the distribution of the error term, contrary to
most other existing approaches in the literature. We propose to penalize the
estimator proposed by Stute for censored linear regression by the l1-norm. We
derive rates of convergence and establish asymptotic normality of the estimator
of the regression coefficients. Our estimator has the same asymptotic variance
as Stute's estimator in the censored linear model without outliers. Hence,
there is no loss of efficiency as a result of robustness. Tests and confidence
sets can therefore rely on the theory developed by Stute. The outlined
procedure is also computationally advantageous, since it amounts to solving a
convex optimization program. We also propose a second estimator which uses the
proposed penalized Stute estimator as a first step to detect outliers. It has
similar theoretical properties but better performance in finite samples as
assessed by simulations.
