The Armitage–Doll model with random frailty can fail to describe incidence rates of rare cancers influenced by an accelerated biological mechanism at some, possibly short, period of life. We propose a new model to account for this influence. Osteosarcoma and Ewing sarcoma are primary bone cancers with characteristic age–incidence patterns that peak in adolescence. We analyze Surveillance, Epidemiology and End Result program incidence data for whites younger than 40 years diagnosed during the period 1975–2005, with an Armitage–Doll model with compound Poisson frailty. A new model treating the adolescent growth spurt as the accelerated mechanism affecting cancer development is a significant improvement over that model. We also model the incidence rate conditioning on the event of having developed the cancers before the age of 40 years and compare the results with those predicted by the Armitage–Doll model. Our results support existing evidence of an underlying susceptibility for the two cancers among a very small proportion of the population. In addition, the modeling results suggest that susceptible individuals with a rapid growth spurt acquire the cancers sooner than they otherwise would have if their growth had been slower. The new model is suitable for modeling incidence rates of rare diseases influenced by an accelerated biological mechanism. Copyright © 2012 John Wiley & Sons, Ltd.