August 29th, 2009 | 
Author: A non negative random variable or its distribution is frequently called a risk. In principle any distribution, concentrated on the non negative half-line, can be used as a claim size distribution. However, we will often make a mental distinction between “well-behaved” distributions and dangerous distributions with a heavy tail. Concepts like well-behaved or heavy-tailed distributions belong to the common vocabulary of actuaries. We will make a serious attempt to formalize them in a mathematically sound definition. Roughly speaking, the class of well-behaved distributions consists of those distributions F with an exponentially bounded tail, for some positive.
The condition means that large claims are not impossible, but the probability of their occurrence decreases exponentially fast to zero as the threshold becomes larger and larger. As we will see in later chapters, this condition enhances the exponential-type behavior of most important actuarial diagnostics like aggregate claim amount and ruin probabilities.
We will give a somewhat streamlined approach to heavy tailed distributions. For such distributions there is no proper exponential bound and huge claims are getting more likely. A natural nonparametric class of heavy-tailed claim size distributions is the class S of sub exponential distributions introduced and studied. The class S has some extremely neat probabilistic properties that will be highlighted whenever possible. For example, the aggregate claim amount is mainly determined by the largest claim in the portfolio. From the practical point of view, however, the class S is too wide since it cannot be characterized by parameters having a useful interpretation. For this reason practitioners usually fall back on “weakly parametrized” sub exponential distributions. For example, the Normalizes distribution belongs to S and is extremely popular in modeling motor insurance claim data. However, for the case of fire or catastrophic event insurance, the Pareto distribution with, seems to be saliently flexible to cope with most practical examples. It is fortunate that, over the past few decades, the asymptotically and statistical properties of sub exponential distributions have received considerable attention. In this textbook we will, however, mainly deal with asymptotic properties and only touch upon some of the resulting statistical issues.
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