N-estimation from retrospectively ascertained events with applications to AIDS

It is a common sampling scheme in retrospective studies that the data set includes only individuals who satisfy a certain sampling criterion. In this paper we consider the situation when the sampling criterion is a specified event, and assume that an earlier event can be retrospectively identified given the occurrence of the specified event. A semiparametric method, which is a compromise between nonparametric and parametric methods, is employed for the estimation of the expected number of the specified events (namely, the N- estimation) occurring in arbitrarily given intervals. A number of statistical properties of the estimates are developed. Due to the limitation of semiparametric models, our estimates should be regarded as conservative estimates since in general they underestimate the actual number of the specified events. This type of limitation, however, cannot be avoided with nonparametric or semiparametric models. Applications to acquired immunodeficiency syndrome (AIDS) cases are considered. The blood transfusion AIDS cases reported to the Centers for Disease Control are analyzed in detail.