The Aalen-Johansen estimator is a multi-state (matrix) version of the Kaplan–Meier estimator for the hazard of a survival process. The estimator can be used to estimate the transition probability matrix of a Markov process with a finite number of states. 
The position in state space for an entity in continuous time is a random variable taking values in the state space S (We assume a finite state space ).
The estimator is given by the expression
is the transition matrix element from time s to time t, the mn-th element of the matrix denotes the probability that the Markov process starting in state m at time s will be in state n at time t. The summation is over all times where transition events are observed (a total of K).
The estimation of the transition intensities at any time where transitions are observed is simply by counting:
where is the number of transitions observed from state m to state n at time and is the number of entities in state m right before time
The diagonal elements are given by
where is the number of transitions away from state n at time
The Aalen-Johansen estimator is equivalent to the cohort method when the latter is applied to very short intervals
- Aalen, O. O. and Johansen, S. (1978). An empirical transition matrix for nonhomogeneous Markov chains based on censored observations. Scandinavian Journal of Statistics 5, 141–150.