  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.
