Asymptotic distributions of statistics and parameter estimates for mixed Poisson processes

Mixed Poisson processes have been used as natural models for events occurring in continuous or discrete time. Our main result is the derivation of the joint asymptotic distributions of statistics, including parameter estimators, computed in different time intervals from data generated by mixed Poisson processes. These distributions can be used, for example, to test the hypothesis about the adequacy of the mixed Poisson process against data. We provide some simulation results and test the model on actual market research data.