A stochastic model of radiation carcinogenesis is proposed that has much in common with the ideas suggested by M. Pike as early as 1966. The model allows us to obtain a parametric family of substochastic-type distributions for the time of tumor latency that provides a description of the rate of tumor development and the number of affected individuals. With this model it is possible to interpret data on tumor incidence in terms of promotion and progression processes. The basic model is developed for a prolonged irradiation at a constant dose rate and includes short-term irradiation as a special case. A limiting form of the latent time distribution for short-term irradiation at high doses is obtained. This distribution arises in the extreme value theory within the random minima framework. An estimate for the rate of convergence to a limiting distribution is given. Based on the proposed latent time distributions, long-term predictions of carcinogenic risk do not call for information about irradiation dose. As shown by computer simulation studies and real data analysis, the parametric estimation of carcinogenic risk appears to be robust to the loss of statistical information caused by the right-hand censoring of time-to-tumor observations. It seems likely that this property, although revealed by means of a purely empirical procedure, may be useful in selecting a model for the practical purpose of risk prediction.