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Asymptotic behavior of the unconditional NPMLE of the length-biased survivor function from right censored prevalent cohort data. (English) Zbl 1086.62113

Summary: Right censored survival data collected on a cohort of prevalent cases with constant incidence are length-biased, and may be used to estimate the length-biased (i.e., prevalent-case) survival function. When the incidence rate is constant, so-called stationarity of the incidence, it is more efficient to use this structure for unconditional statistical inference than to carry out an analysis by conditioning on the observed truncation times. It is well known that, due to the informative censoring for prevalent cohort data, the Kaplan Meier estimator is not the unconditional NPMLE of the length-biased survival function and the asymptotic properties of the NPMLE do not follow from any known result.
We present here a detailed derivation of the asymptotic properties of the NPMLE of the length-biased survival function from right censored prevalent cohort survival data with follow-up. In particular, we show that the NPMLE is uniformly strongly consistent, converges weakly to a Gaussian process, and is asymptotically efficient. One important spin-off from these results is that they yield the asymptotic properties of the NPMLE of the incident-case survival function [see M. Asgharian, C. E. M’Lan and D. B. Wolfson J. Am. Stat. Assoc. 97, No. 457, 201–209 (2002; Zbl 1073.62561)], which is often of prime interest in a prevalent cohort study.
Our results generalize those given by Y. Vardi and C.-H. Zhang [Ann. Stat. 20, No. 2, 1022–1039 (1992; Zbl 0761.62056)] under multiplicative censoring, which we show arises as a degenerate case in a prevalent cohort setting.

MSC:

62N02 Estimation in survival analysis and censored data
62G20 Asymptotic properties of nonparametric inference
62N01 Censored data models
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