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2023 On nonparametric estimation for cross-sectional sampled data under stationarity
Kwun Chuen Gary Chan, Hok Kan Ling, Sheung Chi Phillip Yam
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Electron. J. Statist. 17(2): 2745-2809 (2023). DOI: 10.1214/23-EJS2163


We study the nonparametric estimation of the underlying survival function of a survival time in a study with cross-sectional sampling without any follow-up. Under a stationarity assumption on disease incidence rate in the population, the survival function S0 is related to the observed density of the backward recurrence time, f0, via the relationship S0(x)=f0(x)f0(0). As f0(x) is non-decreasing, it is well-known that the nonparametric maximum likelihood estimator of f0 at x=0 is inconsistent. In this article, we establish the asymptotic distributions of the estimators of S0(x) when different consistent estimators of f0(0) are used. Such results are currently missing in the literature. Another contribution is the establishment of a local Kiefer-Wolfowitz-type result of the form supx[0,y]|Fˆn(x)Fn(x)|=Op(n23(logn)23) that makes use of weaker assumptions than existing results, where Fn and Fˆn are the empirical distribution function and its least concave majorant, respectively.


Gary Chan acknowledges the support by US National Institutes of Health Grant R01HL122212 and US National Science Foundation Grant DMS1711952. Hok Kan Ling thanks for the hospitality at the Chinese University of Hong Kong during the final draft of this work and acknowledges the support by NSERC Grant RGPIN/03124-2021. Phillip Yam acknowledges the financial support from HKGRF – Project Number 14300319 with the project title: Shape-constrained Inference: Testing for Monotonicity. He also thanks Columbia University for the kind invitation to be a visiting faculty member in the Department of Statistics during his sabbatical leave.


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Kwun Chuen Gary Chan. Hok Kan Ling. Sheung Chi Phillip Yam. "On nonparametric estimation for cross-sectional sampled data under stationarity." Electron. J. Statist. 17 (2) 2745 - 2809, 2023.


Received: 1 October 2022; Published: 2023
First available in Project Euclid: 7 November 2023

Digital Object Identifier: 10.1214/23-EJS2163

Primary: 62G07 , 62G20

Keywords: Backward recurrence time , cross-sectional sampling , current duration data , Grenander estimator , Kiefer-Wolfowitze-type result , shape-constrained inference

Vol.17 • No. 2 • 2023
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