The Researches Of Local Linear Smoothers Using The RIG Kernel

This paper mainly discusses in the local linear smoother using asymmetric kernel to estimate the regression curve with bounded support. Some scholars have used the beta and gamma kernel to discuss the local linear smoother and pointes out that either gamma kernels if the curve is bounded from one end only or beta kernels if the curve has a compact support. This paper uses the RIG asymmetric kernels and the asymptotic properties of the RIG kernels is similar to the gamma kernels. It turns out that local linear RIG smoother removes the problem of an increased mean square error near the boundary and the MISE achieves n-4/5order convergence. In addition to the usual good properties of local linear smoothing with a symmetric kernels, the RIG kernel have some extra benefits. Firstly, the RIG kernel has varying shapes and varying amounts of smoothness, so it is a kind of adaptive smoothing. Secondly, when the effective sample size increases, the finite sample variance of the curve estimator can be reduced. This properties is due to the fact that the support of regression curve matches the support of the kernel. When the curve have sparse regions, this can make local linear smoother with a small variance. We could study the local linear smoother used the asymmetric kernels instead of the symmetric kernels. But when the support of the kernel does not match the support of regression curve and the curve has sparse regions, the variance of local linear smoother may be unable to have satisfactory results.This dissertation consists of three chapters. Chapter one is the introduction. In this chapter, we review the origin of the local linear smoother and the results of domestic and foreign scholars. The second chapter discusses the local linear RIG smoother. Firstly, we gives the proof of the Theorem1, the expression of the bias and variance uses the RIG kernel. Secondly, we study the asymptotic properties and give the specific expression of the MSE and its MISE. Obtaining the optimal bandwidth and comparing with the Gamma kernels. The third chapter gives a simulation algorithm, using matlab to make simulated drawing and analyzes the results.