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, 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]
For a natural number and real parameters, with and, the probability mass function of the ExtNegBin( , ) distribution is given by
f(k;m,r,p)=0 fork\in\{0,1,\ldots,m-1\}
and
f(k;m,r,p)=
| {k+r-1\choosek | |
| p |
k}{(1-p)-r
| m-1 | |
| -\sum | |
| j=0 |
{j+r-1\choosej}pj} fork\in{N}withk\gem,
where
{k+r-1\choosek}=
| \Gamma(k+r) | |
| k!\Gamma(r) |
=(-1)k{-r\choosek} (1)
is the (generalized) binomial coefficient and denotes the gamma function.
Using that for is also a probability mass function, it follows that the probability generating function is given by
| infty | |
| \begin{align}\varphi(s)&=\sum | |
| k=m |
| ||||||||||||||||
| f(k;m,r,p)s |
(ps)j} {(1-p)-r
| m-1 | |
| -\sum | |
| j=0 |
\binom{j+r-1}jpj} for|s|\le
| 1p.\end{align} | |
For the important case, hence, this simplifies to
| \varphi(s)= | 1-(1-ps)-r | for|s|\le |
| 1-(1-p)-r |
| 1p. | |