Estimating the Mean of an Increasing Stochastic Process at a Censored Stopping Time

Abstract

Data often come in the form of a longitudinal series, say N(·), whose observation has been “stopped” either by a terminal event U (e.g., death) or censoring event C (e.g., lost to follow-up). When N(·) is not deterministic, it is generally not possible to use standard survival analysis tools (e.g., the Kaplan-Meier estimator) to consistently estimate functionals of the distribution of N(U) because of “induced” informative censoring. Various seemingly unrelated estimators recently have been proposed to handle this difficulty within the context of specific problems, ranging from the analysis of quality-adjusted survival data to recurrent event data to lifetime medical cost data. In this article I consider a general framework containing each of these specific examples as a special case, propose a class of estimators for the mean of the stopped longitudinal process, and establish the asymptotic properties of this class of estimators. Connections between the aforementioned estimators are established, and questions related to asymptotic relative efficiency of the various estimators are addressed. Based on these results, I propose some new estimators with finite-sample optimality properties. Finally, I investigate the utility of the proposed estimators in two simulated examples, and discuss potential extensions of this work.