A new method for proving weak convergence results applied to nonparametric estimators in survival analysis.
The author considers a set of i.i.d. random variables -- survival times, with independent (both on survival times and mutually) random censoring. Unlike in a standard setting, the common distribution function of survival times is allowed to be discontinuous. The asymptotic properties of estimators of the distribution function and of the cumulative hazard rate are studied. Namely, Kaplan-Meier, Nelson-Aalen and Hjort's Bayesian estimators are recalled. It is shown that the relevant counting processes still have a martingale increments structure, so that the weak normality of estimators holds again. This result is proven directly in Part 5 for the Kaplan-Meier and Nelson-Aalen estimator, while the convergence for the Bayes estimator follows from the asymptotic equivalence between Bayes and frequentist estimators, as shown in Part 4. The proof is based on a version of a limit theorem for martingales obtained by \textit{A. Jakubowski} et al. [Probab. Theory Relat. Fields 81, 111-137 (1989; Zbl 0638.60049)]. It is a pity that the author deals with just the simplest case of survival data (i.i.d. random variables), instead with, for instance, a more general case of counting processes defined directly by their intensities, as by \textit{P. K. Andersen} et al. [Statistical models based on counting processes. (1993; Zbl 0769.62061)]. The case of discontinuous cumulated intensities would then lead to multiple counts at one moment and to the mutual dependence of martingale increments.
scientific article; zbMATH DE number 2102355
scientific article