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Censored data are quite common in statistics and have been studied in
depth in the last years [for some references, see Powell (J Econom 25(3):303–325,
1984), Murphy et al. (Math Methods Stat 8(3):407–425, 1999), Chay and Powell
(J Econ Perspect 15(4):29–42, 2001)]. In this paper, we consider censored high-
dimensional data. High-dimensional models are in some way more complex than
their low-dimensional versions, therefore some different techniques are required. For
the linear case, appropriate estimators based on penalised regression have been devel-
oped in the last years [see for example Bickel et al. (Ann Stat 37(4):1705–1732, 2009),
Koltchinskii (Bernoulli 15:799–828, 2009)]. In particular, in sparse contexts, the l 1 -
penalised regression (also known as LASSO) [see Tibshirani (J R Stat Soc Ser B
58:267–288, 1996), Bühlmann and van de Geer (Statistics for high-dimensional data.
Springer, Heidelberg, 2011) and reference therein] performs very well. Only few theo-
retical work was done to analyse censored linear models in a high-dimensional context.
We therefore consider a high-dimensional censored linear model, where the response
variable is left censored. We propose a new estimator, which aims to work with high-
dimensional linear censored data. Theoretical non-asymptotic oracle inequalities are
derived.