A parametric model for heterogeneity in paired Poisson counts

We present a model for data in the form of matched pairs of counts. Our work is motivated by a problem in fission‐track analysis, where the determination of a crystal's age is based on the ratio of counts of spontaneous and induced tracks. It is often reasonable to assume that the counts follow a Poisson distribution, but typically they are overdispersed and there exists a positive correlation between the numbers of spontaneous and induced tracks in the same crystal. We propose a model that allows for both overdispersion and correlation by assuming that the mean densities follow a bivariate Wishart distribution. Our model is quite general, having the usual negative‐binomial and Poisson models as special cases. We propose a maximum‐likelihood estimation method based on a stochastic implementation of the EM algorithm, and we derive the asymptotic standard errors of the parameter estimates. We illustrate the method with a data set of fission‐track counts in matched areas of zircon crystals.