Correcting an estimator of a multivariate monotone function with isotonic regression
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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 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 uniform stochastic equicontinuity condition and that the true function is Lipschitz and strictly monotone. We provide simple sufficient conditions for uniform stochastic equicontinuity in the important special case that the initial estimator is uniformly asymptotically linear. We scrutinize the use of these results for estimation of a G-computed distribution function, a quantity often of interest in causal inference, and a conditional distribution function, and illustrate their implications through numerical results. Our experiments suggest that the projection step can yield practical improvements in performance for both the estimator and confidence band.
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