Bounded length confidence intervals in nonparametric regression

Abstract

Consider the regression model Y=h(x)+e where h is a smooth unknown re,gression function and e has unknown distribution function F, symmetric about 0.Sequential procedures are constructed which provide a confidence interval for h(x) of length at most 2d and at a given point x.We assume that the design points are chosen by the experimenter in such a way that in an asymptotic sense, they are spread out over the interval [0,1] in a uniform way.One procedure is based on regression quantiles, while the other is based on a linear kernel estimator of h(x).Asymptotic theory is developed asd→O and the asymptotic efficiency between the two procedures is derived.

Keywords:

• asymptotic theory
• kernel estimators
• regression quantiles
• sequential estimation