Correcting an estimator of a multivariate monotone function with isotonic regression

Abstract: In many problems, a sensible estimator of a possibly multivariate monotone function may itself fail to be monotone. We study the correction of such an estimator obtained via projection onto the space of functions monotone over a finite grid in the domain. We demonstrate that this corrected estimator has no worse supremal estimation error than the initial estimator, and that analogously corrected confidence bands contain the true function whenever the initial bands do, at no loss to average or maximal band width. Additionally, we demonstrate that the corrected estimator is uniformly asymptotically equivalent to the initial estimator provided that the initial estimator satisfies a stochastic equicontinuity condition and that the true function is Lipschitz and strictly monotone. We provide simple sufficient conditions for our stochastic equicontinuity condition in the important special case that the initial estimator is uniformly asymptotically linear, and illustrate the use of these results for estimation of a G-computed distribution function. Our stochastic equicontinuity condition is weaker than standard uniform stochastic equicontinuity, which has been required for alternative correction procedures. Crucially, this allows us to apply our results to the bivariate correction of the local linear estimator of a conditional distribution function known to be monotone in its conditioning argument. Our experiments suggest that the projection step can yield significant practical improvements in performance for both the estimator and confidence band.