Μεικτές σύνθετες κατανομές Poisson
Mixed compound Poisson distributions
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Keywords
Σύνθετες κατανομές ; Αλγόριθμοι ; Panjer ; Depril ; Poisson processesAbstract
Mixed compound Poisson distributions are used in a range of scientific fields to model
non-homogeneous populations. In this paper, we study the results of the mixed compound
Poisson distributions that often have the desirable properties for the modeling of the claim
distribution and provide interesting actuarial applications. For example, they often have
thick tails which make them useful for long-tailed data. Mixed structural distributions are
given such as (generalized) Gamma, (generalized) Beta, generalized inverse Gaussian, and
the probability functions of the (total) claim distributions are explicitly calculated. We also
present characterizations of mixed Poisson distribution through the absolute monotony of
the probability generating function and the thinning.
In turns out that aggregate claim distributions can be determined explicitly only in
a few exceptional cases. However, the most important claim number distributions can
be characterized by a simple recursion formula and admit the recursive computation of
aggregate claims distributions and their moments, when the claim size distribution is discrete
or continuous.
Finally, detailed proofs of an extension of Panjer’s recursion, due to K.Th. Hess, A.
Liewald and K.D. Schmidt [36], are presented. The results of this extension are of interest
in catastrophe excess-of-loss reinsurance and have an application to Hofmann families