On the role of Volterra integral equations in self-consistent, product-limit, inverse probability of censoring weighted, and redistribution-to-the-right estimators for the survival function.

IF 1.2 3区 数学 Q3 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS
Lifetime Data Analysis Pub Date : 2024-07-01 Epub Date: 2024-03-21 DOI:10.1007/s10985-024-09623-0
Robert L Strawderman, Benjamin R Baer
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引用次数: 0

Abstract

This paper reconsiders several results of historical and current importance to nonparametric estimation of the survival distribution for failure in the presence of right-censored observation times, demonstrating in particular how Volterra integral equations help inter-connect the resulting estimators. The paper begins by considering Efron's self-consistency equation, introduced in a seminal 1967 Berkeley symposium paper. Novel insights provided in the current work include the observations that (i) the self-consistency equation leads directly to an anticipating Volterra integral equation whose solution is given by a product-limit estimator for the censoring survival function; (ii) a definition used in this argument immediately establishes the familiar product-limit estimator for the failure survival function; (iii) the usual Volterra integral equation for the product-limit estimator of the failure survival function leads to an immediate and simple proof that it can be represented as an inverse probability of censoring weighted estimator; (iv) a simple identity characterizes the relationship between natural inverse probability of censoring weighted estimators for the survival and distribution functions of failure; (v) the resulting inverse probability of censoring weighted estimators, attributed to a highly influential 1992 paper of Robins and Rotnitzky, were implicitly introduced in Efron's 1967 paper in its development of the redistribution-to-the-right algorithm. All results developed herein allow for ties between failure and/or censored observations.

论 Volterra 积分方程在生存函数的自洽、乘积限制、反删减概率加权和向右再分布估计器中的作用。
本文重新考虑了在存在右删失观察时间的情况下,对失败的生存分布进行非参数估计的几个具有历史和现实意义的结果,特别展示了 Volterra 积分方程是如何帮助将所得到的估计值相互连接起来的。论文首先考虑了埃夫隆的自洽方程,该方程是在 1967 年伯克利研讨会的一篇开创性论文中提出的。本研究提供的新见解包括:(i) 自洽方程直接导致一个预期伏特拉积分方程,该方程的解由删减生存函数的乘积极限估计器给出;(ii) 本论证中使用的定义立即建立了我们熟悉的失败生存函数的乘积极限估计器;(iii) 失败生存函数的乘积极限估计器的通常伏特拉积分方程立即导致一个简单的证明,即它可以表示为删减加权估计器的反概率;(iv) 一个简单的特征描述了失败生存函数和分布函数的自然删减加权反概率估计器之间的关系;(v) 罗宾斯和罗特尼茨基在 1992 年发表的一篇极具影响力的论文中提出了删减加权反概率估计器,而艾夫隆在 1967 年发表的论文中在发展向右再分配算法时隐含地引入了这一估计器。本文提出的所有结果都考虑了失败和/或删减观测值之间的联系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Lifetime Data Analysis
Lifetime Data Analysis 数学-数学跨学科应用
CiteScore
2.30
自引率
7.70%
发文量
43
审稿时长
3 months
期刊介绍: The objective of Lifetime Data Analysis is to advance and promote statistical science in the various applied fields that deal with lifetime data, including: Actuarial Science – Economics – Engineering Sciences – Environmental Sciences – Management Science – Medicine – Operations Research – Public Health – Social and Behavioral Sciences.
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