Simulation of Posterior Distributions in Nonparametric Censored Analysis

We analyze the model in which the latent durations \(T_i\) are i.i.d. generated by a distribution F . The statistician observes \(Y_i = \min (T_i , C_i)\) and \(A_i = \mathbb {I}_{\{T_i \leq C_i\}}\) where \(C_i\) is a censoring time. The prior probability on F is a Dirichlet process. Hjort (Ann Stat 18(3):1259–1294, 1990) shows that the posterior distribution is a neutral to the right process whose hazard function is a beta process. Lo (Ann Stat 21(1):100–123, 1993) has the same type of results with different assumptions on censoring times. For a large class of specifications on censoring times, we exhibit a representation of the posterior process which has the following form: \(F = \sum _j F_jF^j\) where j indexes the intervals between censoring times, the \(F_j\) ’s are product of independent beta distributed random variables, and the \(F^j\) ’s are independent Dirichlet processes. Using powerful representations of Dirichlet processes (Sethuraman, Stat Sin 4(2):639–650, 1994), we deduce from this property a very efficient way to simulate various functionals of F.