Extended negative binomial distribution
In probability and statistics the extended negative binomial distribution is a discrete probability distribution extending the negative binomial distribution. It is a truncated version of the negative binomial distribution for which estimation methods have been studied. In the context of actuarial science, the distribution appeared in its general form in a paper by K. Hess, A. Liewald and K.D. Schmidt when they characterized all distributions for which the extended Panjer recursion works. For the case , the distribution was already discussed by Willmot and put into a parametrized family with the logarithmic distribution and the negative binomial distribution by H.U. Gerber. Probability mass function

For a natural number  and real parameters ,  with  and , the probability mass function of the ExtNegBin(, , ) distribution is given by

and

where

is the (generalized) binomial coefficient and  denotes the gamma function. Probability generating function
Using that  for  is also a probability mass function, it follows that the probability generating function is given by

For the important case , hence , this simplifies to

References

Discrete distributions
Factorial and binomial topics