Local polynomial M-smoothers in nonparametric regression

Abstract

In the case of the equally spaced fixed design nonparametric regression, the local polynomial M-smoother (LPM) is proposed for estimating the regression function. The LPM is developed directly from both the local constant M-smoother in Chu et al. (J. Amer. Statist. Assoc. 93 (1998) 526) and the local linear M-smoother in Rue et al. (J. Nonparametric Statist. 14 (2002) 155). It is shown that, in the smooth region, our LPM has the same asymptotic bias as the local polynomial estimator, but has larger asymptotic variance. On the other hand, in the jump region, it has the interesting property of edge-preserving, but is inconsistent. In practice, a fast algorithm for computing the LPM is presented, and the idea of cross-validation is applied to select the value of the smoothing parameter. More importantly, the results including the fast computation algorithm obtained for the LPM in the one-dimensional case can be extended directly to the multidimensional case. Simulation studies demonstrate that our LPM is competitive with alternatives, in the sense of yielding both smaller sample mean average squared error and better visual performance.