Estimation of the survival function with increasing failure rate based on left truncated and right censored data

Abstract

SUMMARY The maximum conditional likelihood estimator of the survival function with increasing hazard rate is derived, based on left truncated and right censored data. This estimator always exists, whereas the fully nonparametric conditional maximum likelihood estimator may not exist. A strong consistency theorem is established based on the total time on test transformation. Right-censored survival observations arise naturally in biomedical studies of survival when some of the patients are not followed until death. Parametric and nonparametric methods for analysing such data have been considered by many authors. In the single- sample problem, Kaplan & Meier (1958) derived the nonparametric maximum likelihood estimator of the survival function based on right censored data. In addition to right censoring, data may be subject to left truncation. This arises when the individuals enter the study at some known time after the time origin. For example, in studying the effects on mortality of occupational exposure to agents in a certain industry, age would normally be taken as the primary time variable, but, observation on an individual would not commence until work in this industry started. Specifically, we consider the following. Let X be a random variable representing lifetimes, and let (T, C) be the random variables describing the left truncation times and right censoring times respectively. Suppose that X follows the distribution F with associated survival function S = 1- F We also assume that (T, C) is independent of X and follows the joint distribution G with pr (T < C) = 1. The quantities observed are Y=min(X,C),T and 8, where 8=1 if T-X C and 8=0 if C<X. Nothing is observed if X < T. In effect, observations are made from the distribution of (Y, T, 8)