The NPMLE for Doubly Censored Current Status Data

Abstract

In biostatistical applications interest often focuses on the estimation of the distribution of time T 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, but with its distribution over an observed interval ‘ A, B ’ known to be uniformly distributed; the data is referred to as doubly censored current status data. These authors used this model to handle application in AIDS partner studies focusing on the NPMLE of the distribution G of T . The model is a submodel of the current status model, but the distribution G is essentially the derivative of the distribution of interest F in the current status model. In this paper we establish that the NPMLE of G is uniformly consistent and that the resulting estimators for the n 1/2 ‐estimable parameters are efficient. We propose an iterative weighted pool‐adjacent‐violator‐algorithm to compute the estimator. It is also shown that, without smoothness assumptions, the NPMLE of F converges at rate n −2/5 in L 2 ‐norm while the NPMLE of F in the non‐parametric current status data model converges at rate n −1/3 in L 2 ‐norm, which shows that there is a substantial gain in using the submodel information.