Geometry of exponential family with competing risks and censored data

•The manifold ℱRδ of a statistical model is constructed.•The geometry of the manifold is investigated by using two information sources.•The geometry is employed to expand the predictions and risk evaluations.•An example and prostatic cancer data are analyzed to illustrate main results.

Employing the differential geometrical methods in statistics suggested by Amari (1985) and Amari et al. (1987), considering the exponential family with censored data and competing risks as a manifold of a statistical model, the geometry of the manifold is investigated based on two information sources. As an application of the geometry, the asymptotic expansions of the bootstrap prediction, Bayesian prediction and their risk evaluations are investigated. The results show that these expansions are related to the coefficients of αα-connections and metric tensors, and the predictive density function is the estimative density function in the asymptotic sense. Finally, taking Rayleigh distribution and prostatic cancer data as examples, some computation and simulation results are presented to illustrate our main results.