Asymptotic Distribution Results in Competing Risks Estimation

Abstract

Consider a time-continuous nonhomogeneous Markovian process $V$ having state space $A^0$. For $A \subset A^0$ and $i, j \in A, P_{Aij}(\tau, t)$ is the $i \rightarrow j$ transition probability of the Markovian process $V_A$ which arises in the hypothetical situation where states $A^0 - A$ have been eliminated from the state space of $V$. Let $\hat{P}_{Aij}(\tau, t)$ be the generalized product-limit estimator of $P_{Aij}(\tau,t)$. It is shown that the vector consisting of components in $\{N^\frac{1}{2}(\hat{P}_{Aij}(\tau, t) - P_{Aij}(\tau, t)): i, j \in A; i \neq j\}$ converges weakly to a vector of dependent Gaussian processes. The structure of this limiting vector process is studied. Finally these results are applied to the estimation of certain biometric functions.