A Stochastic Model for Claim Count Data

In actuarial science, weighted and mixed Poisson distributions are widely employed for claim count modeling due to their flexibility in handling over-dispersed data. This study introduces a novel claim count distribution—the Erlang-Poisson (EP) mixture and examines its distinctive properties: (i) a versatile hazard function exhibiting increasing, upside-down bathtub, and V-shaped patterns; (ii) generating function, moments and index of dispersion as well as risk assessment analyses; (iii) characterization theorems for identification; and (iv) parameter estimation via method of moments (MoM) and maximum likelihood estimation (MLE), with simulations favoring MLE. Empirical tests on six insurance datasets demonstrate the EP model’s superiority over traditional distributions (e.g., negative binomial, generalized Poisson, discrete Lindley), validating its efficacy for real-world claim count analysis. Ultimately, the estimated model indicates that claim counts are highly zero-inflated and overdispersed, demonstrating that the model is well-suited for accurately assessing insurance risk and facilitating informed decision-making in claims management. Our findings suggest that this new model is suitable for analyzing claim counts in real-world insurance data. Some conclusions are drawn.