Nonparametric Estimation for Nonhomogeneous Markov Processes in the Problem of Competing Risks

Abstract

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.