Relaxed Poisson cure rate models

The purpose of this article is to make the standard promotion cure rate model (Yakovlev and Tsodikov, [Yakovlev, A., 1996]) more flexible by assuming that the number of lesions or altered cells after a treatment follows a fractional Poisson distribution (Laskin, [Laskin, N., 2003]). It is proved that the well‐known Mittag‐Leffler relaxation function (Berberan‐Santos, [Berberan‐Santos, M., 2005]) is a simple way to obtain a new cure rate model that is a compromise between the promotion and geometric cure rate models allowing for superdispersion. So, the relaxed cure rate model developed here can be considered as a natural and less restrictive extension of the popular Poisson cure rate model at the cost of an additional parameter, but a competitor to negative‐binomial cure rate models (Rodrigues et al., [Rodrigues, J., 2009a]). Some mathematical properties of a proper relaxed Poisson density are explored. A simulation study and an illustration of the proposed cure rate model from the Bayesian point of view are finally presented.