Confidence intervals in monotone regression

TL;DR

This paper develops a consistent bootstrap method for constructing confidence intervals in monotone regression using a smoothed estimator, improving over traditional nonparametric bootstrap approaches.

Contribution

It introduces a smoothed bootstrap approach for monotone regression, providing asymptotic distribution derivations and automatic bandwidth selection, enhancing inference accuracy.

Findings

Smoothed bootstrap yields consistent confidence intervals.

Comparison shows improved performance over non-monotone estimators.

Automatic bandwidth selection method is effective.

Abstract

We construct bootstrap confidence intervals for a monotone regression function. It has been shown that the ordinary nonparametric bootstrap, based on the nonparametric least squares estimator (LSE) $\hat f_n$ is inconsistent in this situation. We show, however, that a consistent bootstrap can be based on the smoothed $\hat f_n$, to be called the SLSE (Smoothed Least Squares Estimator). The asymptotic pointwise distribution of the SLSE is derived. The confidence intervals, based on the smoothed bootstrap, are compared to intervals based on the (not necessarily monotone) Nadaraya Watson estimator and the effect of Studentization is investigated. We also give a method for automatic bandwidth choice, correcting work in Sen and Xu (2015). The procedure is illustrated using a well known dataset related to climate change.