Английская Википедия:Extended negative binomial distribution

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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 Шаблон:Math, 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 Шаблон:Math and real parameters Шаблон:Mvar, Шаблон:Mvar with Шаблон:Math and Шаблон:Math, the probability mass function of the ExtNegBin(Шаблон:Mvar, Шаблон:Mvar, Шаблон:Mvar) distribution is given by

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

and

<math> 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,</math>

where

<math> {k+r-1 \choose k} = \frac{\Gamma(k+r)}{k!\,\Gamma(r)} = (-1)^k\,{-r \choose k}\qquad\qquad(1)</math>

is the (generalized) binomial coefficient and Шаблон:Math denotes the gamma function.

Probability generating function

Using that Шаблон:Math for Шаблон:MathШаблон:Open-closed is also a probability mass function, it follows that the probability generating function is given by

<math>\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}</math>

For the important case Шаблон:Math, hence Шаблон:MathШаблон:Open-open, this simplifies to

<math>

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