Singly and Doubly Censored Current Status Data: Estimation, Asymptotics and Regression

In biostatistical applications interest often focuses on the estimation of the distribution of time between two consecutive events. If the initial event time is observed and the subsequent event time is only known to be larger or smaller than an observed point in time, then the data is described by the well-understood singly censored current status model, also known as interval censored data, case I. Jewell et al. (1994) extended this current status model by allowing the initial time to be unobserved, with its distribution over an observed interval (A, Bl known; the data is referred to as doubly censored current status data. This model has applications in AIDS partner studies. If the initial time is known to be uniformly distributed, the model reduces to a submodel of the current status model with the same asymptotic information bounds as in the current status model, but the distribution of interest is essentially the derivative of the distribution of interest in the current status model. As a consequence the non-parametric maximum likelihood estimator is inconsistent. Moreover, this submodel contains only smooth heavy tailed distributions for which no moments exist. In this paper, we discuss the connection between the singly censored current status model and the doubly censored current status model (for the uniform initial time) in detail and explain the difficulties in estimation which arise in the doubly censored case. We propose a regularized MLE corresponding with the current status model. We prove rate results, efficiency of smooth functionals of the regularized MLE, and present a generally applicable efficient method for estimation of regression parameters, which does not rely on the existence of moments. We also discuss extending these ideas to a non-uniform distribution for the initial time. In many longitudinal studies, interest focuses on the distribution of the length of the interval between two events. This situation is particularly common in epidemiological investigations of the natural history of a disease. Jewell et al. (1994) give two examples that arise from studies of human immunodeficiency virus (HIV) disease. Notationally, let I and J be the chronological times of the two events, respectively, with I - J always. Suppose that, for a given individual, we observe whether either of both of the two events have occurred at a single point in time, B. We assume that only individuals for whom I - B, that is, the first event has occurred by time B, are included in the sample of individuals available for study. We are interested in two forms of data structure; in the first I (- B) is always observed and measurement of J is censored in that we only observe whether J B. For the second situation, measurement of both I and J is limited to observing whether each of these times is greater than or less than B. In both cases we are concerned with non-parametric esti- mation of the distribution G of T = J - I, based on n i.i.d. observations, under the assumption that T is independent of I. The first structure is commonly referred to as (singly censored) current status data. The qualifier "singly censored" serves to differentiate this structure from the second kind discussed in further detail below. Previous work and examples of current status data can be found in Diamond et al. (1986), Jewell & Shiboski (1990), Diamond & McDonald (1991), and Keiding