Extended negative binomial distribution

In probability and statistics the extended negative binomial distribution is a discrete probability distribution extending the negative binomial distribution.

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

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

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.

Proof that the probability mass function is well defined

Note that for all k ≥ m

${\binom {k+r-1}{k}}={\biggl (}\prod _{j=1}^{m}{\frac {j+r-1}{j}}{\biggr )}\prod _{j=m+1}^{k}{\Bigl (}1+{\frac {r-1}{j}}{\Bigr )}$

has the same sign and, using log(1 + x) ≤ x for x > –1 and noting that r – 1 < 0,

${\begin{aligned}\log \prod _{j=m+1}^{k}{\Bigl (}1+{\frac {r-1}{j}}{\Bigr )}&\leq \sum _{j=m+1}^{k}{\frac {r-1}{j}}\\&\leq (r-1)\int _{m+1}^{k+1}{\frac {dx}{x}}=\log {\Bigl (}{\frac {k+1}{m+1}}{\Bigr )}^{r-1}.\end{aligned}}$

Therefore,

$\sum _{k=m}^{\infty }{\biggl |}{\binom {k+r-1}{k}}{\biggr |}\leq {\biggl |}\prod _{j=1}^{m}{\frac {k+r-1}{j}}{\biggr |}\sum _{k=m}^{\infty }{\Bigl (}{\frac {m+1}{k+1}}{\Bigr )}^{1-r}<\infty$

by the integral test for convergence, because 1 – r > 1. Using (1) and Abel's theorem, we see that the binomial series representation

$(1-x)^{-r}=\sum _{k=0}^{\infty }{\binom {-r}{k}}(-x)^{k}$

holds for all x in [–1,1]. Hence, the probability mass functions actually sums up to one.

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

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

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

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