Estimation of extreme survival probabilities with cox model

We propose an extension of the regular Cox's proportional hazards model which allows the estimation of the probabilities of rare events. It is known that when the data are heavily censored, the estimation of the tail of the survival distribution is not reliable. To improve the estimate of the baseline survival function in the range of the largest observed data and to extend it outside, we adjust the tail of the baseline distribution beyond some threshold by an extreme value model under appropriate assumptions. The survival distributions conditioned to the covariates are easily computed from the baseline. A procedure allowing an automatic choice of the threshold and an aggregated estimate of the survival probabilities are also proposed. The performance is studied by simulations and an application on two data sets is given.