A New Method for Proving Weak Convergence Results Applied to Hjort's Nonparametric Bayes Estimators

In 1990, Hjort introduced nonparametric Bayes estimators of the cumulative distribution function and the cumulative hazard rate, based on type I censored data. Our aim in this paper is to study their large sample behaviour. Firstly, we develop a martingale structure for each estimator. Then, we prove that they are almost surely equivalent to the frequentist estimators of Nelson-Aalen and Kaplan-Meier. Finally, a new method for proving weak convergence results, which allows discontinuities for the underlying distribution, is introduced.