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.

给出了回归函数的一种新颖的局部多项式估计。传统的局部多项式估计器使用全局恒定带宽来调整估计的平滑度。具有局部可变带宽的方法，其中一些方法仅通过考虑输入的不均匀分布而不考虑输出的方式来选择带宽，而其他方法通常需要进行一些复杂的预先估计。我们构造了一个自适应局部多项式估计器，它通过使用新的局部变量带宽来考虑输出的变化和输入的排列，并且不需要额外的计算负担。将新估计的渐近偏差，方差和均方误差与传统的局部多项式估计进行比较。