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 of the first kind 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 of the first kind 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 (i.e., under appropriate conditions). Finally, we show that 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.

翻译：本文重新审视了在右删失观测时间下，对生存分布进行非参数估计的历史与当前重要结果，特别展示了第一类沃尔泰拉积分方程如何帮助连接这些估计量。论文首先回顾了Efron在1967年伯克利研讨会经典论文中提出的自洽方程。本研究的新见解包括：(i) 自洽方程直接导出一个第一类预见性沃尔泰拉积分方程，其解由删失生存函数的乘积极限估计量给出；(ii) 该论证中使用的定义立即建立了失效生存函数的经典乘积极限估计量；(iii) 失效生存函数乘积极限估计量的标准沃尔泰拉积分方程可简单直接地证明其在适当条件下能表示为逆删失概率加权估计量。最后，我们证明归功于Robins与Rotnitzky 199年极具影响力论文的逆删失概率加权估计量，实际上已隐含在Efron 1967年论文对右向再分配算法的推导中。本文所有结果均允许失效观测和/或删失观测之间存在结值。