The Bernstein-von Mises theorem for martingale-residual processes in Cox-type regression models

The Cox model is a convenient choice for survival data analysis, however, even though it is a semiparametric model and therefore applicable to many various scenarious, it still has its restrictions. Among them belong the proportional hazard assumption, al l individuals behaving according to the same baseline hazard etc. Goodness of fit testing based on martingale-residual processes provides good way to detect potential defects in model. In special cases the limiting distribution of such processes is a tr ansformed Brownian motion what gives a tool for testing. In general, the asymptotic distribution of the processes is rather tedious to be used directly for testing and either transformation or approximative simulations must be carried out. Recently in Ba yesian survival analysis, a lot of attention was paid to asymptotic properties of infinite-dimensional parameters. In work of several authors it was established that under certain family of priors for the cumulative hazard function the posterior distribut ion of the parameters centered around the maximum likelihood estimators (MLE) is asymptotically equivalent to the sampling distribution of MLE (the Bernstein-von Mises theorem, BvM). Consequently, similar result can be settled for any smooth functional of estimated parameters. Our main interest is to derive the Bayesian asymptotic properties of the generalized martingale-residual process as a functional of parameters. Once we have a BvM-like statement for it, we can base the tests on the posterior distri bution rather then on difficult frequentist asymptotics. We would also like to investigate general versions of Cox model with time-dependent covariates and general relative risk. Further, we will propose various ways of constructing credible bands once a MCMC sample from the posterior distribution is obtained, from simple point-wise sample quantiles to the depth of functional data and regression quantiles. Achieved results will be assessed in a simulations study.