On nonparametric estimation for cross-sectional sampled data under stationarity

Abstract

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 $S_0$ is related to the observed density of the backward recurrence time, $f_0$, via the relationship $S_0(x)=f_0(x)∕f_0(0)$. As $f_0(x)$ is non-decreasing, it is well-known that the nonparametric maximum likelihood estimator of $f_0$ at $x=0$ is inconsistent. In this article, we establish the asymptotic distributions of the estimators of $S_0(x)$ when different consistent estimators of $f_0(0)$ 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 ${sup}_{x\in [0,y]}|{\hat{F}}_n(x)-{\mathbb{F}}_n(x)|=O_p(n^{-2∕3}{(logn)}^{2∕3})$ that makes use of weaker assumptions than existing results, where ${\mathbb{F}}_n$ and ${\hat{F}}_n$ are the empirical distribution function and its least concave majorant, respectively.