Consider a time-continuous nonhomogeneous Markovian stochastic process $V$ having state space $A^0$. Let $A \subset A^0$ and let $P_{Aij}(\tau, t)$ be the $i \rightarrow j$ transition probability of the Markovian stochastic process $V_A$ arising in the hypothetical situation where states $A^0 - A$ have been eliminated from the state space of $V$. Based upon the concept of Kaplan and Meier's product-limit estimator, a nonparametric estimator $\hat{P}_{Aij}(\tau, t)$ is formulated which is proved to be uniformly strongly consistent and asymptotically unbiased. These results generalize those by Aalen for the special case in which $A^0$ has one transient state.
@article{1176344310,
author = {Fleming, Thomas R.},
title = {Nonparametric Estimation for Nonhomogeneous Markov Processes in the Problem of Competing Risks},
journal = {Ann. Statist.},
volume = {6},
number = {1},
year = {1978},
pages = { 1057-1070},
language = {en},
url = {http://dml.mathdoc.fr/item/1176344310}
}
Fleming, Thomas R. Nonparametric Estimation for Nonhomogeneous Markov Processes in the Problem of Competing Risks. Ann. Statist., Tome 6 (1978) no. 1, pp. 1057-1070. http://gdmltest.u-ga.fr/item/1176344310/