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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

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  • Robert L. Strawderman

    (University of Rochester)

  • Benjamin R. Baer

    (University of Rochester
    University of St Andrews)

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.

Suggested Citation

  • Robert L. Strawderman & Benjamin R. Baer, 2024. "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," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 30(3), pages 649-666, July.
    • Handle: RePEc:spr:lifeda:v:30:y:2024:i:3:d:10.1007_s10985-024-09623-0
    DOI: 10.1007/s10985-024-09623-0
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    References listed on IDEAS

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    1. Mangalam, Vasudevan & Nair, Gopalan M. & Zhao, Yun, 2008. "On computation of NPMLE for middle-censored data," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1452-1458, September.
    2. Satten G. A. & Datta S., 2001. "The Kaplan-Meier Estimator as an Inverse-Probability-of-Censoring Weighted Average," The American Statistician, American Statistical Association, vol. 55, pages 207-210, August.
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