A note on uniform consistency of monotone function estimators

Recently, Dette et al. [A simple nonparametric estimator of a strictly increasing regression function. Bernoulli 12, 469–490] proposed a new monotone estimator for strictly increasing nonparametric regression functions and proved asymptotic normality. We explain two modifications of their method that can be used to obtain monotone versions of any nonparametric function estimators, for instance estimators of densities, variance functions or hazard rates. The method is appealing to practitioners because they can use their favorite method of function estimation (kernel smoothing, wavelets, orthogonal series, etc.) and obtain a monotone estimator that inherits desirable properties of the original estimator. In particular, we show that both monotone estimators share the same rates of uniform convergence (almost sure or in probability) as the original estimator.