Strictly monotone and smooth nonparametric regression for two or more variables

The authors propose a new monotone nonparametric estimate for a regression function of two or more variables. Their method consists in applying successively one-dimensional isotonization procedures on an initial, unconstrained nonparametric regression estimate. In the case of a strictly monotone regression function, they show that the new estimate and the initial one are first-order asymptotic equivalent; they also establish asymptotic normality of an appropriate standardization of the new estimate. In addition, they show that if the regression function is not monotone in one of its arguments, the new estimate and the initial one have approximately the same L-p-norm. They illustrate their approach by means of a simulation study, and two data examples are analyzed.