Classical Backfitting for Smooth-Backfitting Additive Models

Smooth backfitting has been shown to have better theoretical properties than classical backfitting for fitting additive models based on local linear regression. In this article, we show that the smooth backfitting procedure in the local linear case can be alternatively performed as a classical backfitting procedure with a different type of smoother matrices. These smoother matrices are symmetric and shrinking and some established results in the literature are readily applicable. The connections allow the smooth backfitting algorithm to be implemented in a much simplified way, give new insights on the differences between the two approaches in the literature, and provide an extension to local polynomial regression. The connections also give rise to a new estimator at data points. Asymptotic properties of general local polynomial smooth backfitting estimates are investigated, allowing for different orders of local polynomials and different bandwidths. Cases of oracle efficiency are discussed. Computer-generated simulations are conducted to demonstrate finite sample behaviors of the methodology and a real data example is given for illustration. Supplementary materials for this article are available online.

已有研究表明，在基于局部线性回归（local linear regression）拟合可加模型（additive models）时，平滑反向拟合（smooth backfitting）相较于经典反向拟合（classical backfitting）具备更优的理论性质。本文证明，局部线性场景下的平滑反向拟合流程，可通过一类采用不同平滑矩阵（smoother matrices）的经典反向拟合流程等价实现。这类平滑矩阵兼具对称性与收缩性，现有文献中的诸多成熟结论可直接复用。二者的关联不仅大幅简化了平滑反向拟合算法的实现流程，还为理解文献中两种方法的差异提供了全新视角，同时将相关结论推广至局部多项式回归（local polynomial regression）场景。该关联还可导出一种针对数据点的新型估计量。本文研究了一般局部多项式平滑反向拟合估计量的渐近性质，该分析允许使用不同阶数的局部多项式与不同带宽（bandwidths）。本文还探讨了先知效率（oracle efficiency）的相关情形。本文通过计算机模拟实验展示了所提方法的有限样本表现，并给出真实数据集示例加以说明。本文的补充材料可在线获取。