(Tartu Ülikool, 2013-06-11) Carsten, Frazier Henry; Käärik, Meelis, juhendaja; Tartu Ülikool. Matemaatika-informaatikateaduskond; Tartu Ülikool. Matemaatilise statistika instituut

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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.