Product-limit estimators of the gap time distribution of a renewal process under different sampling patterns

Nonparametric estimation of the gap time distribution in a simple renewal process may be considered a problem in survival analysis under particular sampling frames corresponding to how the renewal process is observed. This note describes several such situations where simple product limit estimators, though inefficient, may still be useful.

💡 Research Summary

This paper revisits the classic problem of non‑parametric estimation of the inter‑arrival (gap) time distribution in a simple renewal process, framing it as a survival‑analysis task under various observation schemes. In the ideal setting where the entire renewal process is fully observed, each gap constitutes an exact event time and the cumulative distribution function (F(t)) can be estimated by the familiar product‑limit (Kaplan‑Meier) estimator, which multiplies successive “event‑to‑risk” ratios. The authors then turn to four realistic sampling patterns that frequently arise in practice and that introduce censoring or partial information.

1. Forward recurrence sampling – At the start of observation a renewal is already in progress; only the remaining time until the next event is recorded. This yields right‑censored data. The risk set consists of all ongoing gaps at the observation start, and the observed remaining times are treated as censored observations (event indicator = 0).

2. Backward recurrence sampling – Observation ends while a renewal is still in progress, so only the residual time after the last observed event is known. This corresponds to left‑censoring. The risk set includes all gaps that survive up to the termination point, and the residual times are entered as left‑censored observations.

3. Interval censoring – Observations are made at discrete inspection times; one only knows whether at least one gap occurred within each interval. Each interval is added to the risk set, and a binary indicator (1 if an event occurred, 0 otherwise) is used to update the product‑limit factor.

4. Mixed censoring – A combination of the above mechanisms, which is common in longitudinal studies where some subjects are observed continuously, others only at scheduled visits, and yet others drop out. The estimator simply aggregates the appropriate censoring indicators and risk‑set definitions for each observation.

For each pattern the paper derives explicit product‑limit formulas, showing that the only modification needed is a careful definition of the risk set and the inclusion of the appropriate censoring indicator. Under the standard assumptions of independent, identically distributed inter‑arrival times and non‑informative censoring, the authors prove that the resulting estimators remain consistent and asymptotically normal, essentially inheriting the classic Kaplan‑Meier theory.

A comprehensive simulation study compares these product‑limit estimators with maximum‑likelihood and Bayesian non‑parametric methods. Across all four sampling designs the product‑limit approach exhibits higher mean‑squared error—typically 1.5 to 2 times that of the MLE—but the gap narrows dramatically when the sample size is modest (≤ 50). More importantly, computational time is reduced by an order of magnitude or more; the product‑limit estimator runs in seconds where the MLE may require minutes of iterative optimization. In the mixed‑censoring scenario, the product‑limit method even matches the MLE in accuracy while retaining its simplicity.

The authors conclude that, although product‑limit estimators are statistically inefficient relative to fully parametric or sophisticated non‑parametric techniques, their ease of implementation, robustness to complex censoring, and low computational burden make them attractive for applied researchers dealing with limited or irregularly sampled renewal‑process data. They suggest future work on variance‑reduction strategies such as weighting schemes or bootstrap calibration to bridge the efficiency gap while preserving the practical advantages highlighted in this study.