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2020 Correcting an estimator of a multivariate monotone function with isotonic regression
Ted Westling, Mark J. van der Laan, Marco Carone
Electron. J. Statist. 14(2): 3032-3069 (2020). DOI: 10.1214/20-EJS1740


In many problems, a sensible estimator of a possibly multivariate monotone function may 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 band width. Additionally, we demonstrate that the corrected estimator is asymptotically equivalent to the initial estimator if the initial estimator satisfies a stochastic equicontinuity condition and the true function is Lipschitz and strictly monotone. We provide simple sufficient conditions in the special case that the initial estimator is 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. 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.


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Ted Westling. Mark J. van der Laan. Marco Carone. "Correcting an estimator of a multivariate monotone function with isotonic regression." Electron. J. Statist. 14 (2) 3032 - 3069, 2020.


Received: 1 December 2019; Published: 2020
First available in Project Euclid: 15 August 2020

zbMATH: 1448.62056
MathSciNet: MR4135324
Digital Object Identifier: 10.1214/20-EJS1740

Primary: 62G20
Secondary: 60G15

Keywords: Asymptotic linearity , Confidence band , kernel smoothing , projection , shape constraint , stochastic equicontinuity


Vol.14 • No. 2 • 2020
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