Triple smoothing estimation of the regression function and its derivatives in nonparametric regression

Abstract

Abstract In the case of the random design nonparametric regression, to correct for the unbounded finite-sample variance of the local polynomial estimator using a ‘global’ bandwidth, several remedies are available. Such methods include, for example, the locally weighted regression estimator (Cleveland, J. Amer. Statist. Assoc. 74 (1979) 829), the empirical-bias bandwidth selection estimator (Ruppert, J. Amer. Statist. Assoc. 92 (1997) 1049), the local ridge regression estimator (Seifert and Gasser, J. Amer. Statist. Assoc. 91 (1996) 267), and the shrinkage estimator (Hall and Marron, Probab. Theory Related Fields 108 (1997) 495). However, in practice, estimates produced by these four remedies might have rough appearance. This practical drawback is caused by the essence of the local polynomial fit. The smaller the sample size, the more often this drawback occurs. To avoid such drawback, the triple smoothing estimator (TSE) for the rth derivative of the regression function is proposed. Here r⩾0 and the 0th derivative stands for the regression function itself. The TSE uses global smoothing parameters and has advantages in both the finite sample and the asymptotic cases. In the finite sample case, it has bounded conditional (and unconditional) bias and variance. On the other hand, in the asymptotic case, the TSE has the same mean-square error as the local p-degree polynomial estimator using a global bandwidth, when p−r is odd. However, when p−r is even, the former is better than the latter in a minimax sense. Simulation studies demonstrate that the TSE is better than these four remedies, in both senses of having smaller sample mean integrated square error and giving smoother estimates.