On Adaptation to sparse design in bivariate local linear regression
Peter Hall, Burkhardt Seifert, Berwin A. Turlach · 2001
INTRODUCTION Problems of nonparametric regression with multivariate design points arise with increasing frequency in a range of applications, including dimension-reduction methods such as projection pursuit and ACE (e.g. Friedman and Stuetzle 1981, Breiman and Friedman 1985, Huber 1985), flexible multivariate models for high-dimensional data (e.g. Friedman 1988, 1991; Friedman and Silverman 1989), and generalized additive models (e.g. Hastie and Tibshirani 1986, Cleveland and Devlin 1988). In one dimension, the virtues of local linear smoothing are well-known (e.g. Fan 1993, Hastie and Loader 1993). Multivariate generalizations of theoretical properties have been discussed by Ruppert and Wand (1994), in the context of local polynomial smoothing. However, both the attractive features (such as excellent theoretical performance) and the disadvantages (including difficulties with sparse design) of local linear smoothing generalize. Indeed, the "curse of dimensionality" correctly p