| Summary: | While over-dispersed Poisson age-period-cohort and extended chain-ladder 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 two-way 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 quasi-likelihood 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 over-dispersed Poisson model assumes that the over-dispersion 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 log-normal age-period-cohort 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 non-nested test that allows one to evaluate whether the over-dispersed Poisson or log-normal model is the better choice for the data. While the over-dispersed Poisson model imposes a fixed variance to mean ratio, the log-normal 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.
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