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

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

and

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

where

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

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

${\displaystyle {\begin{aligned}\varphi (s)&=\sum _{k=m}^{\infty }f(k;m,r,p)s^{k}\\&={\frac {(1-ps)^{-r}-\sum _{j=0}^{m-1}{\binom {j+r-1}{j}}(ps)^{j}}{(1-p)^{-r}-\sum _{j=0}^{m-1}{\binom {j+r-1}{j}}p^{j}}}\qquad {\text{for }}|s|\leq {\frac {1}{p}}.\end{aligned}}}$

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

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

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 and Economics. 10 (4): 303–309. doi:10.1016/0167-6687(92)90061-F. ISSN 0167-6687. MR 1172687. Zbl 0743.62014.