Nonparametric Kernel Regressionsubject to Monotonicity

We suggest a biased-bootstrap method for monotonising general linear, kernel-type estimators, for example local linear estimators and Nadaraya-Watson estimators. Attributes of our approach include the fact that it produces smooth estimates, that is applicable to a particularly wide range of estimator types, and that it can be employed after the smoothing step has been implemented. Therefore , an experimenter may use his or her favourite kernel estimator, and their favourite bandwidth selector, to construct the basic nonparametric smoother, and then use our technique to render it monotone in a smooth way. Since our method is based on maximising delity to the conventional empirical approach, subject to monotonicity, then if the original kernel smoother were monotone we would not modify it. More generally, we would adjust it by adjoining weights to data values so as to make least possible change, in the sense of a distance measure, subject to imposing the constraint of monotonicity.