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 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 a random variable with an ExtNegBin(m, r, p) distribution is given by

$f(k;m,r,p)=0\qquad \text{ for }k\in\{0,1,\ldots,m-1\}$

and

$f(k;m,r,p) = \frac{{k+r-1 \choose k} (1-p)^k}{p^{-r}-\sum_{j=0}^{m-1}{j+r-1 \choose j} (1-p)^j}\quad\text{for }k\in{\mathbb N}\text{ with }k\ge m,$

where

${k+r-1 \choose k} = \frac{\Gamma(k+r)}{k!\,\Gamma(r)} = (-1)^k\,{-r \choose k}\qquad\qquad(1)$

is the (generalized) binomial coefficient and Γ denotes the gamma function.

Probability generating function

Using the above binomial series representation and the abbreviation q = 1 − p, it follows that the probability generating function is given by

$\begin{align}\varphi(s)&=\sum_{k=m}^\infty f(k;m,r,p)s^k\\ &=\frac{(1-qs)^{-r}-\sum_{j=0}^{m-1}\binom{j+r-1}j (qs)^j} {p^{-r}-\sum_{j=0}^{m-1}\binom{j+r-1}j q^j} \qquad\text{for } |s|\le\frac1q.\end{align}$

For the important case m = 1, hence r in (–1,0), this simplifies to

$\varphi(s)=\frac{1-(1-qs)^{-r}}{1-p^{-r}} \qquad\text{for }|s|\le\frac1q.$