ON STUDY OF A MODIFIED LOCAL CONSTANT $M$-SMOOTHER

Abstract

In the case of the equally spaced fixed design nonparametric regression, the local constant $M$-smoother (LCM) with local maximizing is proposed by Chu, Glad, Godtliebsen, and Marron (1998) to correct for the effect of discontinuity on the kernel regression estimator. It has the interesting property of jump-preserving. However, in the jump region, it is inconsistent when the magnitude of the noise is larger than the size of the jump in the regression. To adjust for this drawback to the ordinary LCM, we propose to construct the LCM with global maximizing instead of local maximizing as well as with binned data instead of original data. Our proposed estimator is analyzed by the asymptotic mean square error. Both binning and global maximizing have no effect on the asymptotic mean square error of the ordinary LCM in the smooth region, but have an effect on improving the inconsistency of the ordinary LCM in the jump region. Simulation studies demonstrate that the regression function estimate produced by our modified LCM is better than those by alternatives, in the sense of yielding smaller sample mean integrated square error, showing more accurately the location of jump point, and having smoother appearance.