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.
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.
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