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Credibility evaluation for the exponential dispersion family. (English) Zbl 0927.62109

Summary: It has long been established that under regularity conditions, the linear credibility formula with an appropriate credibility factor produces exact fair premium for claims or losses whose distribution is a member of the natural exponential family. Recently, this result has been extended to a richer family of distributions, the exponential dispersion family which comprised of several distributions, some of which are heavy-tailed and as such could be of significant relevance to actuarial science. The family draws its richness from a dispersion parameter \(\sigma^2=1/\lambda\) which is equal to 1 in the case of the natural exponentially family. In this paper neither \(\lambda\) is regarded known, nor a fully specified prior distribution for \(\lambda\) is assumed. Instead, by establishing a link between the m.s.e of the linear credibility and Fisher information we derive optimal credibility for the case where only the mean and variance of \(\lambda\) are specified.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
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