Comparisons between the Local Polynomial Kernel estimators and the Cubic Smoothing Splines

Abstract

The term Nonparametric regression comes to signify the absence of parameters in the regression model. The scope of nonparametric regression is very broad nowadays, ranging from "smoothing" the relationship between two variables in a scatter plot to multiple-regression analysis. Extraordinary, only several years ago, methods of nonparametric regression analysis have been used widely because of advances in statistical techniques and related computing facilities, and are now a serious alternative to many traditional parametric regression modeling. This paper aims to provide using two methods of the nonparametric regression, namely; the Local Polynomial Kernel (LPK) estimator and Cubic Smoothing Splines (CSS) estimator with emphasis on the role of selecting the smoothing parameter (bandwidth) in each estimator. We have investigated the performance of these two smoothing estimators (LLK and CSS) under two different settings, fixed or random designs. This was achieved through conducting many simulation studies using four different example regression functions; two different distributions of errors term either normal or mixture, and different sample sizes. Two study the estimation of the derivative functions using those two smoothing estimators (Local Quadratic/Cubic Kernel and Cubic Smoothing Splines) in the process of smoothing the 1st / 2nd derivative functions respectively also been considered through conducting several simulation studies as well. Applications on real datasets also been considered. Lastly, the advantages and disadvantages of these two nonparametric regression estimators have been discussed.