A Bernstein–von Mises Theorem for Parametric Competing Risks Under Hybrid Censoring

Abstract

We establish a Bernstein–von Mises (BvM) theorem for parametric competing-risks models under hybrid Type-I censoring, where observation stops at the random time τn=min(X(r),T0). Using the counting-process martingale framework, we first prove the local asymptotic normality (LAN) of the model and identify the limiting Fisher information as a block-diagonal matrix composed of operational (τ∗-truncated) cause-specific informations. Unlike previous work, we derive the testing-function (Hellinger-affinity) condition required for posterior tail control from the standard regularity assumptions rather than imposing it as an extra hypothesis. The posterior distribution of n(θ−θ^n) is shown to converge in total variation to a Gaussian law with covariance I(θ0)−1 for every prior positive and continuous at θ0. The convergence rate is OP((logn)3/2n−1/2); a fourth-order smoothness condition removes the logarithmic factor. The abstract conditions are verified for the exponential, Weibull, and Gompertz families, and a simulation study corroborates the asymptotic approximation and the nominal coverage of Bayesian credible sets.