Lifetime Data Analysis, 2, 119-129 (1996) © 1996 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Estimation from Current-Status Data in Continuous Time NIELS KEIDING University of Copenhagen, Denmark KAMILLA BEGTRUP University of Copenhagen, Denmark THOMAS H. SCHEIKE University of Copenhagen, Denmark GUNTHER HAS!BEDER Technical University of Vienna, Austria Received December 14, 1995; accepted February 26, 1996 Abstract. The nonparametric maximum likelihood estimator for current-status data has been known for at least 40 years, but only recently have the mathematical-statisticalproperties been clarified. This note provides a case study in the important and often studied context of estimating age-specific immunization intensities from a seroprevalence survey. Fully parametric and spline-basedalternatives(also based on continuous-time models) are given. The basic reproduction number Ro exemplifiesestimation of a functional. The limitations implied by the necessarily rather restrictive epidemiologicalassumptions are briefly discussed. Keywords: Age-specific incidence, basic reproduction number, epidemiology, smoothing splines, Weibull sur- vival distribution. 1. Introduction Age-specific immunization rates are basic building blocks in any detailed epidemiological study of diseases with life-long immunity (measles, mumps, rubella, hepatitis). Direct estimation of these rates would be available from a follow-up study, in which a population of susceptibles would be followed for a period of time with all infections recorded, but this survey design is unfortunately rarely feasible. It is much more common to have a cross-sectional sample where for each person, current age and current immunization status is known. In the survival analysis context such data may be said to be all censored, either to the right (if immunization has not yet happened) or to the left (if the person is immune). There has been considerable recent interest in the analysis of such current-status data, see Jewell and Shiboski (1990), Diamond and McDonald (1992), Shiboski and Jewell (1992), Grummer-Strawn (1993), Sun and Kalbfleisch (1993), Andersen and Ronn (1995), Rabinowitz et al. (1995), Jewell and van der Laan (1995) and Rossini and Tsiatis (1996). In this note we assume continuous time and focus on the nonparametric maximum likelihood estimator (NPMLE) first derived by Ayer et al. (1955). Early mathematical-statistical studies of the large-sample properties of such estimators were by Prakasa Rao (1969) and