  As its name suggests, sufficient dimension reduction (SDR) targets to
estimate a subspace from data that contains all information sufficient to
explain a dependent variable. Ample approaches exist to SDR, some of the most
recent of which rely on minimal to no model assumptions. These are defined
according to an optimization criterion that maximizes a nonparametric measure
of association. The original estimators are nonsparse, which means that all
variables contribute to the model. However, in many practical applications, an
SDR technique may be called for that is sparse and as such, intrinsically
performs sufficient variable selection (SVS). This paper examines how such a
sparse SDR estimator can be constructed. Three variants are investigated,
depending on different measures of association: distance covariance, martingale
difference divergence and ball covariance. A simulation study shows that each
of these estimators can achieve correct variable selection in highly nonlinear
contexts, yet are sensitive to outliers and computationally intensive. The
study sheds light on the subtle differences between the methods. Two examples
illustrate how these new estimators can be applied in practice, with a slight
preference for the option based on martingale difference divergence in the
bioinformatics example.
