# 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[1] for which estimation methods have been studied.[2]

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[3] when they characterized all distributions for which the extended Panjer recursion works. For the case m = 1, the distribution was already discussed by Willmot[4] and put into a parametrized family with the logarithmic distribution and the negative binomial distribution by H.U. Gerber.[5]

## 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

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

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 ( . ; m, r, ps) for s (0, 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, 0), this simplifies to

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

## References

1. ^ Jonhnson, N.L.; Kotz, S.; Kemp, A.W. (1993) Univariate Discrete Distributions, 2nd edition, Wiley ISBN 0-471-54897-9 (page 227)
2. ^ Shah S.M. (1971) "The displaced negative binomial distribution", Bulletin of the Calcutta Statistical Association, 20, 143–152
3. ^ Hess, Klaus Th.; Anett Liewald; Klaus D. Schmidt (2002). "An extension of Panjer's recursion" (PDF). ASTIN Bulletin 32 (2): 283–297. doi:10.2143/AST.32.2.1030. MR 1942940. Zbl 1098.91540.
4. ^ Willmot, Gordon (1988). "Sundt and Jewell's family of discrete distributions" (PDF). ASTIN Bulletin 18 (1): 17–29. doi:10.2143/AST.18.1.2014957.
5. ^ Gerber, Hans U. (1992). "From the generalized gamma to the generalized negative binomial distribution". Insurance: Mathematics amd Economics 10 (4): 303–309. doi:10.1016/0167-6687(92)90061-F. ISSN 0167-6687. MR 1172687. Zbl 0743.62014.