  The multivariate adaptive regression spline (MARS) is one of the popular
estimation methods for nonparametric multivariate regressions. However, as MARS
is based on marginal splines, to incorporate interactions of covariates,
products of the marginal splines must be used, which leads to an unmanageable
number of basis functions when the order of interaction is high and results in
low estimation efficiency. In this paper, we improve the performance of MARS by
using linear combinations of the covariates which achieve sufficient dimension
reduction. The special basis functions of MARS facilitate calculation of
gradients of the regression function, and estimation of the linear combinations
is obtained via eigen-analysis of the outer-product of the gradients. Under
some technical conditions, the asymptotic theory is established for the
proposed estimation method. Numerical studies including both simulation and
empirical applications show its effectiveness in dimension reduction and
improvement over MARS and other commonly-used nonparametric methods in
regression estimation and prediction.
