标题： 用等距回归校正多元单调函数的估计量 显示英文标题

标题： Correcting an estimator of a multivariate monotone function with isotonic regression

摘要： 在许多问题中，一个可能的多元单调函数的合理估计量本身可能不是单调的。我们研究了通过在域上有限网格上的单调函数空间上投影得到的该估计量的修正方法。我们证明，这种修正后的估计量的 supremal 估计误差不会比初始估计量更差，并且类似地修正后的置信带在初始带包含真实函数时也包含真实函数，而不会影响平均或最大带宽。此外，我们证明，如果初始估计量满足随机等度连续条件，并且真实函数是 Lipschitz 并且严格单调的，则修正后的估计量与初始估计量在整体渐近意义上等价。我们在初始估计量是整体渐近线性的重要特殊情况下，提供了我们随机等度连续条件的简单充分条件，并说明了这些结果在 G-计算分布函数估计中的应用。我们的随机等度连续条件比标准的整体随机等度连续条件要弱，后者是其他修正程序所需要的。关键的是，这使我们能够将我们的结果应用于已知在其条件变量中单调的条件分布函数的局部线性估计量的二元修正。我们的实验表明，投影步骤可以显著提高估计量和置信带的性能。 显示英文摘要

摘要： 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.