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### Annett Keller

##### ISBN 978-3-8325-2037-3 132 Seiten, Erscheinungsjahr: 2008 Preis: 34.00 €
An important branch in insurance mathematics is the pricing of possible large claims that are either the results of many small claims occuring at once or that are caused by single events.

A premium calculation principle that emphasises the structure of an insurance portfolio is the so called credibility premium.The credibility premium is a convex combination of the class mean, representing the insurance portfolio's general behaviour and the individual mean. The latter takes into account the individual claim history of the risks subsumed in the portfolio. The insurer calculating the premium does not necessarily need to know the claim amount distribution, even though she has to make some assumptions.

In this thesis an insurance portfolio of $N$ risks -- then called risk classes -- is considered. It is assumed that each of the risks typically causesa small claim amount during an insurance period. But once in a while, the risks may produce large claim amounts due to a contamination of the small claim amount distribution function.

For such models to calculate an insurance premium, the credibility approach can be applied combined with methods from robust statistics. In that case, both the claim amounts and the insurance premiums are separated into ordinary and extreme parts. The premium for the ordinary part is determined by applying the credibility principle. We assume the claim amount distribution function of risk $i,\,i=1,\ldots,N$ to be $\Gamma(\alpha,\theta_i)$ with risk parameter $\theta_i$, being a random variable itself. The distribution function of the independent risk parameters $\theta_i$ is known. The rare, large claim amounts originate from a contamination of the claim amount distribution function $\Gamma(\alpha,\theta_i)$. Thus, we will introduce robust estimators. Determining the premium of the extreme part, the mean excess function is going to be used.

After a brief introduction of conecpts in robust statistics, such as robust M-estimators and influence functions, we will define two robust scale M-estimators with respect to our data model, both of them depending on parameters $a$ and $b$. We also discuss the question of choosing optimal values for $a$ and $b$.

Besides we are going to compute the influence functions, gross errors and finite sample breakdown points for these estimators. It is also proved that the two estimators are asymptotically normally distributed.

The thesis is completed by a simulation study. We analyse the sensitivity of the robust scale M-estimators towards different choices of $a$ and $b$, as well as changing sample sizes and possible occurings of large claims. The simulation will show that for reasonable choices of $a$ and $b$, the robust estimators can bear the comparison with the median, which is known as the most robust estimator. As well, we will estimate the credibility premiums for an insurance portfolio consisting of 25 risk classes and discuss the circumstances, when an actuary should apply the robust credibility approach.

Keywords:
• Insurance Mathematics
• M-Estimator
• Robust Estimator
• Large Claim Insurance

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