A UNIFIED STUDY OF NONPARAMETRIC INFERENCE FOR MONOTONE FUNCTIONS
A UNIFIED STUDY OF NONPARAMETRIC INFERENCE FOR MONOTONE FUNCTIONS
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DOI：
10.1214/19-aos1835
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发表时间：
2020-04-01
影响因子：
4.5
通讯作者：
Carone, Marco
中科院分区：
文献类型：
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作者：
Westling, Ted;Carone, Marco
单调函数的非参数推断问题在许多特殊情况下得到了广泛的研究。估计考虑往往是所谓的Grenander型，被表示为左导数的最大凸次要或最小凹优估计的原始函数。本文给出了单调函数的一类广义Grenander型估计在分布上相合和逐点收敛的一般条件。这个广泛的类允许在域的数据依赖转换上执行最小化或最大化操作，可能在实践中产生好处。此外，我们提供了更简单的条件和更具体的分布理论的重要情况下，原始估计和数据相关的转换函数是渐近线性的。我们使用我们的一般结果的背景下，各种研究问题，并表明，我们很容易恢复古典的结果分别建立在每种情况下。更重要的是，我们表明，我们的研究结果使我们能够解决更具有挑战性的问题，涉及参数的使用灵活的学习策略似乎是必要的。特别是，我们研究的单调密度和风险函数的推理使用信息右删失数据，扩展经典的工作独立删失，协变量边缘化的条件均值函数，扩展经典的工作单调回归函数。
The problem of nonparametric inference on a monotone function has been extensively studied in many particular cases. Estimators considered have often been of so-called Grenander type, being representable as the left derivative of the greatest convex minorant or least concave majorant of an estimator of a primitive function. In this paper, we provide general conditions for consistency and pointwise convergence in distribution of a class of generalized Grenander-type estimators of a monotone function. This broad class allows the minorization or majoratization operation to be performed on a data-dependent transformation of the domain, possibly yielding benefits in practice. Additionally, we provide simpler conditions and more concrete distributional theory in the important case that the primitive estimator and data-dependent transformation function are asymptotically linear. We use our general results in the context of various well-studied problems, and show that we readily recover classical results established separately in each case. More importantly, we show that our results allow us to tackle more challenging problems involving parameters for which the use of flexible learning strategies appears necessary. In particular, we study inference on monotone density and hazard functions using informatively right-censored data, extending the classical work on independent censoring, and on a covariate-marginalized conditional mean function, extending the classical work on monotone regression functions.