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

Consider a time-continuous nonhomogeneous Markovian stochastic process V having state space A⁰. Let A \⊂ A⁰ and let PAij(\τ, t) be the i \→ j transition probability of the Markovian stochastic process VA arising in the hypothetical situation where states A⁰ - 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 ̂Aij(\τ, 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 A⁰ has one transient state.