A Non-iterative Estimator for Interval Sampling and Doubly Truncated Data

Abstract

Interval sampling is often used in Survival Analysis and reliability studies. With interval sampling, the sampling information is restricted to the lifetimes of the individuals or units who fail between two specific dates $d_0$ and $d_1$. Thus, this sampling procedure results in randomly doubly truncated data, where the (possibly negative) left-truncation variable is the time from onset to $d_0$, and the right-truncation variable is the left-truncation variable plus the interval width $d_1-d_0$. In this setting, the nonparametric maximum likelihood estimator (NPMLE) of the lifetime distribution is the Efron–Petrosian estimator, a non-explicit estimator which must be computed in an iterative way. In this paper we introduce a non-iterative, nonparametric estimator of the lifetime distribution and we investigate its performance relative to that of the NPMLE. Simulation studies and illustrative examples are provided. The main conclusion of this piece of work is that the non-iterative estimator, being much simpler, performs satisfactorily. Application of the proposed estimator for general forms of random double truncation is discussed.