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 m = 1 , 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]
Probability mass function [ edit ] 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
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{\displaystyle f(k;m,r,p)=0\qquad {\text{ for ))k\in \{0,1,\ldots ,m-1\))
and
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{\displaystyle 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\geq m,}
where
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{\displaystyle {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 .
Probability generating function [ edit ] 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
φ
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{\displaystyle {\begin{aligned}\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|\leq {\frac {1}{p)).\end{aligned))}
For the important case m = 1 , hence r ∈ (–1, 0) , this simplifies to
φ
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{\displaystyle \varphi (s)={\frac {1-(1-ps)^{-r)){1-(1-p)^{-r))}\qquad {\text{for ))|s|\leq {\frac {1}{p)).}
^ Jonhnson, N.L.; Kotz, S.; Kemp, A.W. (1993) Univariate Discrete Distributions , 2nd edition, Wiley ISBN 0-471-54897-9 (page 227)
^ Shah S.M. (1971) "The displaced negative binomial distribution", Bulletin of the Calcutta Statistical Association , 20, 143–152
^ Hess, Klaus Th.; Anett Liewald; Klaus D. Schmidt (2002). "An extension of Panjer's recursion" (PDF) . ASTIN Bulletin . 32 (2): 283–297. doi :10.2143/AST.32.2.1030 . MR 1942940 . Zbl 1098.91540 .
^ Willmot, Gordon (1988). "Sundt and Jewell's family of discrete distributions" (PDF) . ASTIN Bulletin . 18 (1): 17–29. doi :10.2143/AST.18.1.2014957 .
^ Gerber, Hans U. (1992). "From the generalized gamma to the generalized negative binomial distribution". Insurance: Mathematics and Economics . 10 (4): 303–309. doi :10.1016/0167-6687(92)90061-F . ISSN 0167-6687 . MR 1172687 . Zbl 0743.62014 .
Discrete univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint) Directional Degenerate and singular Families