Inconsistency of the MLE for the Joint Distribution of Interval-Censored Survival Times and Continuous Marks

This paper considers the non-parametric maximum likelihood estimator (MLE) for the joint distribution function of an interval-censored survival time and a continuous mark variable. We provide a new explicit formula for the MLE in this problem. We use this formula and the mark-specific cumulative hazard function of Huang & Louis (1998) to obtain the almost sure limit of the MLE. This result leads to necessary and sufficient conditions for consistency of the MLE, which imply that the MLE is inconsistent in general. We show that the inconsistency can be repaired by discretizing the marks. Our theoretical results are supported by simulations. Copyright (c) 2007 Board of the Foundation of the Scandinavian Journal of Statistics..