Adaptive local polynomial estimations for heterogeneously variational regression functions

Abstract

A novel local polynomial estimator for regression functions is given. Traditional local polynomial estimators adjust the smoothness of estimations using a global constant bandwidth. The methods with local variable bandwidths, some of them select the bandwidths by considering only the inhomogeneous distribution of inputs but not outputs, and the others usually require some complicated pre-estimations. We construct an adaptive local polynomial estimator, which considers both the variations in the outputs and the arrangements of the inputs by using a new local variable bandwidth and needs no additional computational burden. The asymptotic bias, variance and mean squared error of the new estimations are compared with traditional local polynomial estimations. Simulation work shows that the new method outperforms the traditional local polynomial estimator and some other methods in estimating heterogeneously variational regression functions.