Nonparametric Estimation under Censoring and Passive Registration

The classical random censorship model assumes that we follow an individual continuously up to the time of failure or censoring, so observing this time as well as the indicator of its type. Under passive registration we only get information on the state of the individual at random observation or registration times. In this paper we assume that these registration times are the times of events in an independent Poisson process, stopped at failure or censoring; the time of failure is also observed if not censored. This problem turns up in historical demography, where the survival time of interest is the life‐length, censoring is by emigration, and the observation times are times of births of children, and other life‐events. (Church registers contain dates of births, marriages, deaths, but not emigrations.) The model is shown to be related to the problem of estimating a density known to be monotone. This leads to an explicit description of the non‐parametric maximum likelihood estimator of the survival function (based on i.i.d. observations from this model) and to an analysis of its large sample properties.