While overdispersed Poisson ageperiodcohort and extended chainladder models are used in a number of fields, so far no rigorous statistical theory has been available. We consider models for aggregate data organized in a twoway table with age and cohort as indices, but without measures of exposure. In these models, used for example in actuarial science, demography, economics, epidemiology and sociology, the number of parameters grows with the number of observations. Thus, standard asymptotic theory is invalid. In Chapter 2, we propose a repetitive structure that keeps the dimension of the table fixed while increasing the latent exposure. We pair this with the assumptions of infinitely divisible distributions which include a variety of compound Poisson models and Poisson mixture models. We then show that Poisson quasilikelihood estimation results in asymptotic t parameter distributions, F inference, and t forecast distributions. In Chapter 3, we build on the asymptotic framework from Chapter 2 and develop tests for model specification. The overdispersed Poisson model assumes that the overdispersion is common across the data. A further assumption is that effects do not have breaks, for example age effects do not vary over cohorts. A lognormal ageperiodcohort model makes similar assumptions. We show that these assumptions can easily be tested and that similar tests can be used in both models. In Chapter 4, we develop a nonnested test that allows one to evaluate whether the overdispersed Poisson or lognormal model is the better choice for the data. While the overdispersed Poisson model imposes a fixed variance to mean ratio, the lognormal models assumes the same for the standard deviation to mean ratio. We leverage this insight to propose a test that has high power to distinguish between the two models. Again, the theory is asymptotic but does not build on a large size of the array and instead makes use of information accumulating within the cells.
