On credibility evaluation and the tail area of the exponential dispersion family

It has recently been established that the credibility formula for claim distributions belonging to the exponential dispersion family with a known dispersion parameter λ provides the exact predicted mean for a future claim. This paper addresses the case when λ is unknown and when its prior distribution is unspecified. An "optimal" credibility formula is derived for the case when one can specify the probability that claims exceed a given threshold, corresponding to the likelihood of very large claims.