Alternative representations of the Aalen-Johansen estimator of the state occupation probabilities in multi-state models

Multi-state models are a flexible framework for analysing processes that evolve through different states over time, with applications in medicine, demography, and economics, among others. A key quantity of interest is the state occupation probability, commonly estimated by the Aalen-Johansen estimator. In this article, we provide alternative representations of this estimator by extending both the redistribution-to-the-right algorithm and the inverse probability of censoring weighted (IPCW) empirical average representation to the general multi-state setting. We show that these formulations are mathematically equivalent to the Aalen-Johansen estimator, offering new interpretative insights and practical understanding.