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

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{{cite journal

| first = Klaus Th.

| last = Hess

|author2=Anett Liewald|author3=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

| doi = 10.2143/AST.32.2.1030

| mr = 1942940 | zbl = 1098.91540

| doi-access = free

}} 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{{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

| doi = 10.2143/AST.18.1.2014957

| doi-access = free

}} and put into a parametrized family with the logarithmic distribution and the negative binomial distribution by H.U. Gerber.{{cite journal

| first = Hans U.

| last = Gerber

| year = 1992

| title = From the generalized gamma to the generalized negative binomial distribution

| journal = Insurance: Mathematics and Economics

| volume = 10

| issue = 4

| pages = 303–309

| issn = 0167-6687

| doi = 10.1016/0167-6687(92)90061-F

| mr = 1172687 | zbl = 0743.62014

}}

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}}, {{mvar|r}}, {{mvar|p}}) distribution is given by

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

and

: $f(k;m,r,p) = \frac{{k+r-1 \choose k} p^k}{(1-p)^{-r}-\sum_{j=0}^{m-1}{j+r-1 \choose j} p^j}\quad\text{for }k\in{\mathbb N}\text{ with }k\ge m,$

where

: ${k+r-1 \choose k} = \frac{\Gamma(k+r)}{k!\,\Gamma(r)} = (-1)^k\,{-r \choose k}\qquad\qquad(1)$

is the (generalized) binomial coefficient and {{math|Γ}} denotes the gamma function.

Probability generating function

Using that {{math|f ( . ; m, r, ps)}} for {{math|s ∈ }}{{open-closed|0, 1}} is also a probability mass function, it follows that the probability generating function is given by

: $\begin{align}\varphi(s)&=\sum_{k=m}^\infty f(k;m,r,p)s^k\\
&=\frac{(1-ps)^{-r}-\sum_{j=0}^{m-1}\binom{j+r-1}j (ps)^j}
{(1-p)^{-r}-\sum_{j=0}^{m-1}\binom{j+r-1}j p^j}
\qquad\text{for } |s|\le\frac1p.\end{align}$

For the important case {{math|m {{=}} 1}}, hence {{math|r ∈ }}{{open-open|–1, 0}}, this simplifies to

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

References

Category:Discrete distributions

Category:Factorial and binomial topics