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, r, p) distribution is given by

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

and

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

where

$(\genfrac{}{}{0ex}{}{k+r-1}{k})=\frac{\Gamma (k+r)}{k!\Gamma (r)}=(-1{)}^k(\genfrac{}{}{0ex}{}{-r}{k})(1)$

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

$\begin{array}{cc}\phi (s)& =\sum _{k=m}^{\mathrm{\infty }}f(k;m,r,p)s^k\\ & =\frac{(1-ps{)}^{-r}-\sum _{j=0}^{m-1}(\genfrac{}{}{0ex}{}{j+r-1}{j})(ps{)}^j}{(1-p{)}^{-r}-\sum _{j=0}^{m-1}(\genfrac{}{}{0ex}{}{j+r-1}{j})p^j}\text{for }|s|\le \frac{1}{p}.\end{array}$

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

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