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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 distribution|truncated]] version of the negative binomial distribution<ref>Jonhnson, N.L.; Kotz, S.; Kemp, A.W. (1993) ''Univariate Discrete Distributions'', 2nd edition, Wiley ISBN 0-471-54897-9 (page 227)</ref> for which estimation methods have been studied.<ref>Shah S.M. (1971) "The displaced negative binomial distribution", ''Bulletin of the Calcutta Statistical Association'', 20, 143–152</ref> 

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<ref name="Schmidt">{{cite journal
| first = Klaus Th.
| last = Hess
| coauthors = Anett Liewald, Klaus D. Schmidt
| year = 2002
| title = An extension of Panjer's recursion
| journal = ASTIN Bulletin
| volume = 32
| issue = 2
| pages = 283–297
| url = http://www.casact.org/library/astin/vol32no2/283.pdf
| format=PDF
| doi = 10.2143/AST.32.2.1030
| mr = 1942940 | zbl = 1098.91540
}}</ref> when they characterized all distributions for which the extended [[Panjer recursion]] works. For the case {{math|''m'' {{=}} 1}}, the distribution was already discussed by Willmot<ref name="Willmot">{{cite journal
| first = Gordon
| last = Willmot
| year = 1988
| title = Sundt and Jewell's family of discrete distributions
| journal = ASTIN Bulletin
| volume = 18
| issue = 1
| pages = 17–29
| url = http://www.casact.org/library/astin/vol18no1/17.pdf
| format=PDF
| doi = 10.2143/AST.18.1.2014957
}}</ref> and put into a parametrized family with the [[logarithmic distribution]] and the negative binomial distribution by H.U. Gerber.<ref name="Gerber">{{cite journal
| first = Hans U.
| last = Gerber
| year = 1992
| title = From the generalized gamma to the generalized negative binomial distribution
| journal = Insurance: Mathematics amd Economics
| volume = 10
| issue = 4
| pages = 303–309
| issn = 0167-6687
| doi = 10.1016/0167-6687(92)90061-F
| mr = 1172687 | zbl = 0743.62014
}}</ref>

==Probability mass function==

For a natural number {{math|''m'' ≥ 1}} and real parameters {{mvar|p}}, {{mvar|r}} with {{math|0 < ''p'' ≤ 1}} and {{math|–''m'' < ''r'' < –''m'' + 1}}, the [[probability mass function]] of the ExtNegBin({{mvar|m}},&thinsp;{{mvar|r}},&thinsp;{{mvar|p}}) 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 {{math|Γ}} denotes the [[gamma function]].

==Probability generating function==
Using that {{math|''f''&thinsp;(&thinsp;.&thinsp;;&thinsp;''m'',&thinsp;''r'',&thinsp;''ps'')}} for {{math|''s'' ∈ }}{{open-closed|0,&thinsp;1}} 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 {{math|''m'' {{=}} 1}}, hence {{math|''r'' ∈ }}{{open-open|–1,&thinsp;0}}, this simplifies to

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

==References==
<references />

{{ProbDistributions|discrete-infinite}}

[[Category:Discrete distributions]]
[[Category:Factorial and binomial topics]]
[[Category:Probability distributions]]