Overdispersed models for claim count distribution
Constructing models to predict future loss events is a fundamental duty of actuaries. However, large amounts of information are needed to derive such a model. When considering many similar data points (e.g., similar insurance policies or individual claims), it is reasonable to create a collective risk model, which deals with all of these policies/claims together, rather than treating each one separately. By forming a collective risk model, it is possible to assess the expected activity of each individual policy. This information can then be used to calculate premiums (see, e.g., Gray & Pitts, 2012). There are several classical models that are commonly used to model the number of claims in a given time period. This thesis is primarily concerned with the Poisson model, but will also consider the Negative Binomial model and, to a lesser extent, the Binomial model. We will derive properties for each of these models, both in the case when all insurance policies cover the same time period, and when they cover different time periods. The primary focus of this thesis is overdispersion, which occurs when the observed variance of the data in a model is greater than would be expected, given the model parameters. We consider several possible treatments for overdispersion, particularly those that apply to the Poisson model. First, we attempt to generalize the Poisson model by adding an overdispersion parameter (see, e.g., K¨a¨arik & Kaasik, 2012). Next, we search for ways to convert an overdispersed Poisson model to a Negative Binomial model. We will derive some basic properties (such as expectation, variance, and additivity properties) for all of the models mentioned above. Finally, results of this thesis are explored in a practical sense, by attempting to fit computer-generated data into an overdispersed Poisson framework.