  Dimension reduction and variable selection are performed routinely in
case-control studies, but the literature on the theoretical aspects of the
resulting estimates is scarce. We bring our contribution to this literature by
studying estimators obtained via L1 penalized likelihood optimization. We show
that the optimizers of the L1 penalized retrospective likelihood coincide with
the optimizers of the L1 penalized prospective likelihood. This extends the
results of Prentice and Pyke (1979), obtained for non-regularized likelihoods.
We establish both the sup-norm consistency of the odds ratio, after model
selection, and the consistency of subset selection of our estimators. The
novelty of our theoretical results consists in the study of these properties
under the case-control sampling scheme. Our results hold for selection
performed over a large collection of candidate variables, with cardinality
allowed to depend and be greater than the sample size. We complement our
theoretical results with a novel approach of determining data driven tuning
parameters, based on the bisection method. The resulting procedure offers
significant computational savings when compared with grid search based methods.
All our numerical experiments support strongly our theoretical findings.
