Monotonicity preservation properties of kernel regression estimators

TL;DR

This paper analyzes the monotonicity preservation properties of three kernel regression estimators, revealing conditions under which they preserve monotonicity, with implications for their application and kernel choice.

Contribution

It provides a theoretical comparison of the monotonicity preservation properties of NW, PC, and GM kernel estimators, identifying specific conditions on kernels.

Findings

GM estimator always preserves monotonicity.

NW estimator preserves monotonicity if and only if the kernel is log concave.

PC estimator does not preserve monotonicity for any kernel.

Abstract

Three common classes of kernel regression estimators are considered: the Nadaraya--Watson (NW) estimator, the Priestley--Chao (PC) estimator, and the Gasser--M\"uller (GM) estimator. It is shown that (i) the GM estimator has a certain monotonicity preservation property for any kernel $K$, (ii) the NW estimator has this property if and only the kernel $K$ is log concave, and (iii) the PC estimator does not have this property for any kernel $K$. Other related properties of these regression estimators are discussed.