  In this paper, we propose a novel method to select significant variables and
estimate the corresponding coefficients in multiple-index models with a group
structure. All existing approaches for single-index models cannot be extended
directly to handle this issue with several indices. This method integrates a
popularly used shrinkage penalty such as LASSO with the group-wise minimum
average variance estimation. It is capable of simultaneous dimension reduction
and variable selection, while incorporating the group structure in predictors.
Interestingly, the proposed estimator with the LASSO penalty then behaves like
an estimator with an adaptive LASSO penalty. The estimator achieves consistency
of variable selection without sacrificing the root-$n$ consistency of basis
estimation. Simulation studies and a real-data example illustrate the
effectiveness and efficiency of the new method.
