Singly and Doubly Censored Current Status Data: Estimation, Asymptotics and Regression
In biostatistical applications interest is often focused 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, Malani, Vittinghoff (1994) extended this current status model by allowing the initial time to be unobserved, with its distribution over an observed interval [A,B] 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 nonparametric 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 beween 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.
van der Laan, Mark J.; Bickel, Peter J.; and Jewell, Nicholas P., "Singly and Doubly Censored Current Status Data: Estimation, Asymptotics and Regression" (November 1994). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 50.