Classical Backfitting for Smooth-backfitting Additive Models

Abstract

Smooth backfitting has been shown to have better theoretical properties than classical backfitting for fitting additive models based on local linear regression. In this paper we show that the smooth backfitting procedure based on local polynomial regression can be expressed as a classical backfitting procedure with smoother matrices in Huang and Chen (2008). These smoother matrices are symmetric and shrinking and some established results in Buja et al. (1989) are readily applicable. The connections allow the smooth backfitting algorithm to be understood in a much simplified way and give new insights on the differences between the two approaches in the literature. The connections also give rise to a new estimator at data points, which is different from the estimator at a grid point studied in the literature. Asymptotic properties are investigated for the new estimator in the case of bivariate additive models, allowing for different orders of local polynomials. Simulations are conducted to demonstrate finite sample behaviors of the methodology. This is a joint work with Chung-Hsin Yu.