  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.
