Nonparametric estimation of the conditional distribution at regression boundary points

TL;DR

This paper introduces a simple correction method for local linear distribution estimators in nonparametric regression, ensuring monotonicity and improving estimation at boundary points, with demonstrated effectiveness through simulations and real data.

Contribution

A novel correction technique for local linear distribution estimators that guarantees monotonicity at boundary points in nonparametric regression.

Findings

The corrected estimator is monotone increasing.

The method performs well in simulations.

Real data examples confirm effectiveness.

Abstract

Nonparametric regression is a standard statistical tool with increased importance in the Big Data era. Boundary points pose additional difficulties but local polynomial regression can be used to alleviate them. Local linear regression, for example, is easy to implement and performs quite well both at interior as well as boundary points. Estimating the conditional distribution function and/or the quantile function at a given regressor point is immediate via standard kernel methods but problems ensue if local linear methods are to be used. In particular, the distribution function estimator is not guaranteed to be monotone increasing, and the quantile curves can "cross". In the paper at hand, a simple method of correcting the local linear distribution estimator for monotonicity is proposed, and its good performance is demonstrated via simulations and real data examples.