Bayesian nonparametric estimation in the current status continuous mark model

Abstract

We consider the current status continuous mark model where, if an event takes place before an inspection time T a "continuous mark" variable is observed as well. A Bayesian nonparametric method is introduced for estimating the distribution function of the joint distribution of the event time (X) and mark variable (Y ). We consider two histogram-type priors on the density of (X, Y ). Our main result shows that under appropriate conditions, the posterior distribution function contracts pointwisely at rate ( n∕ log n 𝜌+2) if the true density is 𝜌-Hölder continuous. In addition to our theoretical results we provide efficient computational methods for drawing from the posterior relying on a noncentered parameterization and Crank-Nicolson updates. The performance of the proposed methods is illustrated in several numerical experiments.