Nonparametric inference and uniqueness for periodically observed progressive disease models

In many studies examining the progression of HIV and other chronic diseases, subjects are periodically monitored to assess their progression through disease states. This gives rise to a specific type of panel data which have been termed "chain-of-events data"; e.g. data that result from periodic observation of a progressive disease process whose states occur in a prescribed order and where state transitions are not observable. Using a discrete time semi-Markov model, we develop an algorithm for nonparametric estimation of the distribution functions of sojourn times in a J state progressive disease model. Issues of uniqueness for chain-of-events data are not well-understood. Thus, a main goal of this paper is to determine the uniqueness of the nonparametric estimators of the distribution functions of sojourn times within states. We develop sufficient conditions for uniqueness of the nonparametric maximum likelihood estimator, including situations where some but not all of its components are unique. We illustrate the methods with three examples.