A modification of Peto's nonparametric estimation of survival curves for interval-censored data

Peto (1973, Applied Statistics, 22, 86-91) gave a nonparametric generalized maximum-likelihood estimate of the survival function for interval-censored data. His method has a tendency to concentrate probability masses at the endpoints of the intervals, even for the ordinary grouped data, instead of spreading them through the intervals, as one might expect them to be in the underlying distribution. We describe a modification that overcomes this. The new estimate reduces to the standard binomial estimate when applied to grouped data. It also reduces to the Kaplan-Meier estimate when applied to survival data that consist of only exact or right-censored observations. Both estimates are maximum-likelihood estimates but are based on different interpretations of the endpoints of the intervals.