Sequential Fixed-width Confidence Bands for Kernel Regression Estimation

TL;DR: In this article, a fixed-width confidence interval for nonparametric regression with data-driven bandwidths was developed using both Nadaraya-Watson and local linear kernel estimators.

Abstract: We consider a random design model based on independent and identically distributed (iid) pairs of observations (Xi,Yi), where the regression function m(x) is given by m(x) = E(Yi|Xi = x) with one independent variable. In a nonparametric setting the aim is to produce a reasonable approximation to the unknown function m(x) when we have no precise information about the form of the true density, f(x) of X. We describe an estimation procedure of non- parametric regression model at a given point by some appropriately constructed fixed-width (2d) confidence interval with the confidence coefficient of at least1 �. Here, d(> 0) and � 2 (0, 1) are two preassigned values. Fixed-width confidence intervals are developed using both Nadaraya-Watson and local linear kernel esti- mators of nonparametric regression with data-driven bandwidths. The sample size was optimized using the purely and two-stage sequential procedure together with asymptotic properties of the Nadaraya-Watson and local linear estimators. A large scale simulation study was performed to compare their coverage accu- racy. The numerical results indicate that the confi- dence bands based on the local linear estimator have the best performance than those constructed by using Nadaraya-Watson estimator. However both estima- tors are shown to have asymptotically correct cover- age properties.