Nonparametric estimation of a delay distribution based on left-censored and right-truncated data

This paper proposes a nonparametric method for estimating a delay distribution based on left-censored and right-truncated data. A variance-covariance estimator is provided. The method is applied to the Australian AIDS data in which some data are left censored and some data are not left censored. This situation arises with AIDS case-reporting data in Australia because reporting delays were recorded only from November 1990 rather than from the beginning of the epidemic there. It is shown that inclusion of the left-censored data, as opposed to analyzing only the uncensored data, improves the precision of the estimate.     

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.