Asymptotic Distribution Results in Competing Risks Estimation

Consider a time-continuous nonhomogeneous Markovian process V having state space A⁰. For A ⊂ A⁰ and i, j ∈ A, PAij(τ, t) is the i → j transition probability of the Markovian process VA which arises in the hypothetical situation where states A⁰ - A have been eliminated from the state space of V. Let P̂Aij(τ, t) be the generalized product-limit estimator of PAij(τ,t). It is shown that the vector consisting of components in ¹/2(P̂Aij(τ, t) - PAij(τ, t)): i, j ∈ A; i ≠ 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.