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 the ExtNegBin($m$,&thinsp;$r$,&thinsp;$p$) distribution is given by


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

and


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

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 that $f&thinsp;(&thinsp;.&thinsp;;&thinsp;m,&thinsp;r,&thinsp;ps)$ for $s ∈$$(0,&thinsp;1]$ is also a probability mass function, 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-ps)^{-r}-\sum_{j=0}^{m-1}\binom{j+r-1}j (ps)^j} {(1-p)^{-r}-\sum_{j=0}^{m-1}\binom{j+r-1}j p^j} \qquad\text{for } |s|\le\frac1p.\end{align}$$

For the important case $m = 1$, hence $r ∈$$(–1,&thinsp;0)$, this simplifies to



\varphi(s)=\frac{1-(1-ps)^{-r}}{1-(1-p)^{-r}} \qquad\text{for }|s|\le\frac1p.$$