On Bootstrap Confidence Intervals in Nonparametric Regression

Abstract

Several authors have developed bootstrap methods for constructing confidence intervals in nonparametric regression. On each occasion a nonpivotal approach has been employed. Nonpivotal methods are still the overwhelmingly popular choice when statisticians use the bootstrap to compute confidence intervals, but they are not necessarily the most appropriate. In this paper we point out some of the theoretical advantages of pivoting. They include a reduction in the error of the bootstrap distribution function estimate, from $n^{-1/2}$ to $n^{-1}h^{-1/2}$ (where $h$ denotes bandwidth); and a reduction in coverage error of confidence intervals, from either $n^{-1/2}h^{-1/2}$ or $n^{-1/2}h^{1/2}$ (depending on which nonpivotal method is used) to $n^{-1}$. Several comparisons are drawn with the case of nonparametric density estimation, where a pivotal approach also reduces errors associated with confidence intervals, but where the orders of magnitude of the respective errors are quite different from their counterparts for nonparametric regression.