NONPARAMETRIC KERNEL REGRESSION SUBJECT TO MONOTONICITY CONSTRAINTS


       We suggest a method for monotonising 
general kernel-type estimators, for example local linear estimators 
and Nadaraya-Watson estimators.  Attributes of our approach include 
the fact that it produces smooth estimates, indeed with the same 
smoothness as the unconstrained estimate.  The method is applicable 
to a particularly wide range of estimator types, it can be trivially 
modified to render an estimator strictly monotone, and 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.  Implementation involves only an 
off-the-shelf programming routine.  The method is based on maximising 
fidelity to the conventional empirical approach, subject to 
monotonicity.  We adjust the unconstrained estimator by tilting the 
empirical distribution so as to make least possible change, in the 
sense of a distance measure, subject to imposing the constraint of 
monotonicity.  


This is joint work with Professor Peter Hall at Australian National
University.