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 distributionJonhnson, N.L.; Kotz, S.; Kemp, A.W. (1993) Univariate Discrete Distributions, 2nd edition, Wiley (page 227) for which estimation methods have been studied.Shah S.M. (1971) "The displaced negative binomial distribution", Calcutta Statistical Association Bulletin, 20, 143–152
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