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 m = 1, 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 m ≥ 1 and real parameters p, r with 0 < p ≤ 1 and –m < r < –m + 1, the probability mass function of the ExtNegBin(m, r, p) distribution is given by
is the (generalized) binomial coefficient and Γ denotes the gamma function.
Probability generating function
Using that f ( . ; m, r, ps) for s ∈ (0, 1] is also a probability mass function, it follows that the probability generating function is given by
For the important case m = 1, hence r ∈ (–1, 0), this simplifies to