Posterior contraction rate of the proportional hazards model having a nonparametric link and its applications

Abstract

The proportional hazards model was proposed by Cox (1972, 1975), and the theoretical properties of it have been studied widely. Many practical methods have been developed to assess the effects of covariates to the hazard. Bayesian methods for the proportional hazards model can be used very flexible ways. However, the verification of theoretical properties of it is a difficult work. But Kim and Lee (2003) showed that Bayesian methods work well under appropriate priors. The main goal of this thesis is a verification of Bayesian methods using only the partial likelihood and priors. The result is that it can be done through data-dependent priors. In addition, the posterior contraction rates are optimal up to logarithm terms when we use nonparametric functions in the exponents of the proportional hazards models. And these results are extended to the time-varying coefficient models. For an application, we construct a Bayesian test to be performed for proportionality, which is one of the important problems in the survival analysis. The results are promising to develop Bayesian methods only using the partial likelihood, and it is extended to the random effects model and etc. Also, it remains as future works that the more efficient priors or MCMC algorithms.