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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.

A unified study of nonparametric inference for monotone functions