Extended negative binomial distribution

In probability and statistics the extended negative binomial distribution is a discrete probability distribution extending the negative binomial distribution.

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
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] when they characterized all distributions for which the extended Panjer recursion works. For the case "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
format=PDF]

Probability mass function

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 a random variable with an ExtNegBin("m", "r", "p") distribution is given by

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

and

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

where

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

is the (generalized) binomial coefficient and Γ denotes the gamma function.

Proof that the probability mass function is well defined

Note that for all "k" ≥ "m"

:

has the same sign and, using log(1 + "x") ≤ "x" for "x" > –1 and noting that "r" – 1 < 0,

:

Therefore,

:
leiggl|prod_{j=1}^mfrac{k+r-1}jiggr
sum_{k=m}^inftyBigl(frac{m+1}{k+1}Bigr)^{1-r}

by the integral test for convergence, because 1 – "r" > 1. Using (1) and Abel's theorem,we see that the binomial series representation

:

holds for all "x" in [–1,1] . Hence, the probability mass functions actually sums up to one.

Probability generating function

Using the above binomial series representation and the abbreviation "q" = 1 − "p", it follows that the probability generating function is given by

:

For the important case "m" = 1, hence "r" in (–1,0), this simplifies to

:$varphi(s)=frac\{1-(1-qs)^\{-r${1-p^{-rqquad ext{for }|s|lefrac1q.

References