Open Access
April 2020 A unified study of nonparametric inference for monotone functions
Ted Westling, Marco Carone
Ann. Statist. 48(2): 1001-1024 (April 2020). DOI: 10.1214/19-AOS1835


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.


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Ted Westling. Marco Carone. "A unified study of nonparametric inference for monotone functions." Ann. Statist. 48 (2) 1001 - 1024, April 2020.


Received: 1 June 2018; Revised: 1 February 2019; Published: April 2020
First available in Project Euclid: 26 May 2020

zbMATH: 07241578
MathSciNet: MR4102685
Digital Object Identifier: 10.1214/19-AOS1835

Primary: 62G20
Secondary: 62G07 , 62G08 , 62N02

Keywords: cube-root asymptotics , dependent censoring , g-formula , isotonic regression

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 2 • April 2020
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