Nonparametric kernel regression subject to monotonicity constraints

We suggest a method for monotonizing general kernel-type estimators, for ex ample local linear estimators and Nadaraya-Watson estimators. Attributes of our approach include the fact that it produces smooth estimates, indeed wi th the same smoothness as the unconstrained estimate. The method is applica ble to a particularly wide range of estimator types, it can be trivially mo dified to render an estimator strictly monotone and it can be employed afte r the smoothing step has been implemented. Therefore, an experimenter may u se his or her favorite kernel estimator, and their favorite bandwidth selec tor, to construct the basic nonparametric smoother and then use our techniq ue to render it monotone in a smooth way. Implementation involves only an o ff-the-shelf programming routine. The method is based on maximizing fidelit y to the conventional empirical approach, subject to monotonicity. We adjus t the unconstrained estimator by tilting the empirical distribution so as t o make the least possible change, in the sense of a distance measure, subje ct to imposing the constraint of monotonicity.