Abstract
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.
Highlights
Is monotone for each y, in which case θ0 is a bivariate component-wise monotone function
We discuss correcting an initial estimator of a multivariate monotone function by computing the isotonic regression of the estimator over a finite grid in the domain, and interpolating between grid points
Building on the results of Robertson, Wright and Dykstra (1988) and Chernozhukov, Fernandez-Val and Galichon (2009), we demonstrate that the corrected estimator is at least as good as the initial estimator, meaning: (a) its uniform error over the grid used in defining the projection is less than or equal to that of the initial estimator for every sample; (b) its uniform error over the entire domain is less than or equal to that of the initial estimator asymptotically; (c) the corrected confidence band contains the true function on the projection grid whenever the initial band does, at no cost in terms of average or uniform band width
Summary
We discuss correcting an initial estimator of a multivariate monotone function by computing the isotonic regression of the estimator over a finite grid in the domain, and interpolating between grid points. We consider correcting an initial confidence band by using the same procedure applied to the upper and lower limits of the band. (a) its uniform error over the grid used in defining the projection is less than or equal to that of the initial estimator for every sample;. (b) its uniform error over the entire domain is less than or equal to that of the initial estimator asymptotically;. (c) the corrected confidence band contains the true function on the projection grid whenever the initial band does, at no cost in terms of average or uniform band width. 2. We provide high-level sufficient conditions under which the uniform difference between the initial and corrected estimators is oP (rn−1) for a generic sequence rn → ∞
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