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.

翻译：多元自适应回归样条（MARS）是非参数多元回归中常用的估计方法之一。然而，由于MARS基于边际样条，为纳入协变量的交互效应，必须使用边际样条的乘积，这导致交互阶数较高时基函数数量难以控制，并造成估计效率低下。本文通过利用实现充分维度缩减的协变量线性组合来改进MARS的性能。MARS特有的基函数便于计算回归函数的梯度，进而通过梯度外积的特征分析来估计线性组合。在若干技术条件下，建立了所提估计方法的渐近理论。包含模拟与实证应用的数值研究表明，该方法在维度缩减方面具有有效性，并且在回归估计与预测中优于MARS及其他常用非参数方法。