A new method for proving weak convergence results applied to nonparametric estimators in survival analysis.
type: Item
instance of: scholarly article
title: A new method for proving weak convergence results applied to nonparametric estimators in survival analysis.
zbMATH Open document ID: 1046.62047
DOI: 10.1016/S0304-4149(00)00052-1
author: Jean-Yves Dauxois
published in: Stochastic Processes and their Applications
publication date: 2004-09-22T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62N02
Mathematics Subject Classification ID: 62G07
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 62N01
Mathematics Subject Classification ID: 62G05
zbMATH DE Number: 2102355
zbMATH Keywords: stochastic integral
zbMATH Keywords: counting process
zbMATH Keywords: martingale
zbMATH Keywords: weak convergence
zbMATH Keywords: censored data
reviewed by: Petr Volf
MaRDI profile type: Publication
cites work: Nonparametric inference for a family of counting processes
cites work: Statistical models based on counting processes
cites work: Q3999494
cites work: A survey of product-integration with a view toward application in survival analysis
cites work: Nonparametric Bayes estimators based on beta processes in models for life history data
cites work: Q3774629
cites work: Q1099492
cites work: Nonparametric Estimation from Incomplete Observations
cites work: Q4000186
cites work: Weak limit theorems for stochastic integrals and stochastic differential equations
cites work: Q3395931
full work available at URL: s0304-4149(00)00052-1
OpenAlex ID: W2139947932
Recommended article: Q5711197
Recommended article: Q3999494
Recommended article: Convergence des processus de Nelson-Aalen et de Kaplan-Meier par une méthode de martingale
Recommended article: A central limit theorem for functionals of the Kaplan-Meier estimator
Recommended article: Q3094161
Recommended article: Randomization in survival analysis
description: scientific article; zbMATH DE number 2102355