Finite Sum Evaluation of the Negative Binomial-Exponential Model*

Harry H. Panjer and Gordon E. Willmot

ASTIN Bulletin, 1981, vol. 12, issue 2, 133-137

Abstract: The compound negative binomial distribution with exponential claim amounts (severity) distribution is shown to be equivalent to a compound binomial distribution with exponential claim amounts (severity) with a different parameter. As a result of this, the distribution function and net stop-loss premiums for the Negative Binomial-Exponential model can be calculated exactly as finite sums if the negative binomial parameter α is a positive integer.The result is a generalization of Lundberg (1940).Consider the distribution ofwhere X1, X2, X3, … are independently and identically distributed random variables with common exponential distribution functionand N is an integer valued random variable with probability functionThen the distribution function of S is given byIf MX(t), MN(t) and MS(t) are the associated moment generating functions, then

Date: 1981
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