Local Linear Smoothers Using Lognormal Kernel And Birnbaum-Saunders Kernel

This paper considers using the asymmetric kernel functions in local linear s-moothing to estimate a regression curve on[0,?).Since the symmetric kernel function has a boundary effect when estimating the unknown density function,that is,there is a large deviation at the boundary.Such a phenomenon also appears in the regression estimation.Therefore,scholars use the asymmetric kernels to im-prove the problem.In this paper,the asymmetric kernels we want to use are the LN(Lognormal)and the BS(Birnbaum-Saunders)kernel functions.Both LN and BS kernel estimators have non-negative support set and have flexible form and location on the nonnegative real line.The kernel shapes are allowed to vary according to the position of the data points,thus changing the degree of smoothing in a natural way,and their support matches the support of the probability density function to be estimated.This is an adaptive smoothing.Through theoretical derivation,we show the deviation,variance,mean square error and mean integrated square error based on LN and BS kernel in the lagre sample.Compared with the local linear smoothing properties based on RIG and Gamma kernels,LN and BS kernel estima-tors reduces the boundary effect and the deviation at the boundary is smaller than the RIG and Gamma kernel estimators,and achieve the optimal rate of convergence for the mean integrated square error.Through a simulation experiment,we compare the methods of several nonparametric kernel estimates under finite samples,we can find that LN and BS kernel estimators perform better near the boundary in terms of bias reduction under the finite sample,and the MISE of the LN and BS kernels are better than the RIG and Gamma kernels.