Statistical Analysis of Survival Models with Bayesian Bootstrap

Abstract

This chapter presents statistical analysis of survival models with Bayesian bootstrap (BB). The BB was introduced as a Bayesian alternative to the bootstrap and received much attention from both Bayesians and frequentists as tools to approximate the posterior distribution and the sampling distribution. The Rubin's BB is derived in three different approaches, each of which gives a different perspective to the BB. The BB posterior is asymptotically equivalent to the sampling distribution of the maximum likelihood estimator, and often is computationally much simpler than standard frequentist's methods. Moreover, in some cases, the small sample frequentist property of the BB posterior is better than the standard frequentist methods. It is found that as the bootstrap distribution approximates the sampling distributions of a statistic, the Bayesian bootstrap approximates the posterior distribution. Furthermore, in the construction of the Bayesian bootstrap posterior, no prior information is necessary, hinting that the Bayesian bootstrap may be used as noninformative or default nonparametric Bayesian analysis. Among the three views, the empirical likelihood view is adopted and the BB procedure is derived for right censored data, proportional hazard model and doubly censored data.