  An integral part of many algorithms for S-estimators of linear regression is
random subsampling. For problems with only continuous predictors simple random
subsampling is a reliable method to generate initial coefficient estimates that
can then be further refined. For data with categorical predictors, however,
random subsampling often does not work, thus limiting the use of an otherwise
fine estimator. This also makes the choice of estimator for robust linear
regression dependent on the type of predictors, which is an unnecessary
nuisance in practice. For data with categorical predictors random subsampling
often generates singular subsamples. Since these subsamples cannot be used to
calculate coefficient estimates, they have to be discarded. This makes random
subsampling slow, especially if some levels of categorical predictors have low
frequency, and renders the algorithms infeasible for such problems. This paper
introduces an improved subsampling algorithm that only generates nonsingular
subsamples. We call it nonsingular subsampling. For data with continuous
variables it is as fast as simple random subsampling but much faster for data
with categorical predictors. This is achieved by using a modified LU
decomposition algorithm that combines the generation of a sample and the
solving of the least squares problem.
