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 is related to the observed density of the backward recurrence time, , via the relationship . As is non-decreasing, it is well-known that the nonparametric maximum likelihood estimator of at is inconsistent. In this article, we establish the asymptotic distributions of the estimators of when different consistent estimators of 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 that makes use of weaker assumptions than existing results, where and 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.
"On nonparametric estimation for cross-sectional sampled data under stationarity." Electron. J. Statist. 17 (2) 2745 - 2809, 2023. https://doi.org/10.1214/23-EJS2163