Extended negative binomial distribution

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) = frack+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"

:inom{k+r-1}k=iggl(prod_{j=1}^mfrac{j+r-1}jiggr)prod_{j=m+1}^kBigl(1+frac{r-1}jBigr)

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

:egin{align}logprod_{j=m+1}^kBigl(1+frac{r-1}jBigr)&lesum_{j=m+1}^kfrac{r-1}j\&le(r-1)int_{m+1}^{k+1}frac{dx}x=logBigl(frac{k+1}{m+1}Bigr)^{r-1}.end{align}

Therefore,

:sum_{k=m}^inftyiggl|inom{k+r-1}kiggr
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

:(1-x)^{-r}=sum_{k=0}^inftyinom{-r}k(-x)^k

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

:egin{align}varphi(s)&=sum_{k=m}^infty f(k;m,r,p)s^k\&=frac{(1-qs)^{-r}-sum_{j=0}^{m-1}inom{j+r-1}j (qs)^j}{p^{-r}-sum_{j=0}^{m-1}inom{j+r-1}j q^j}qquad ext{for } |s|lefrac1q.end{align}

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


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