Extended negative binomial distribution

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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[1] when they characterized all distributions for which the extended Panjer recursion works. For the case m = 1, the distribution was already discussed by Willmot.[2]

[edit] 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)=0\qquad \text{ for }k\in\{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\text{for }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 Γ denotes the gamma function.

To see that it is a well defined probability distribution, note that for all k ≥ m

\binom{k+r-1}k =\biggl(\prod_{j=1}^m\frac{j+r-1}j\biggr) \prod_{j=m+1}^k\Bigl(1+\frac{r-1}j\Bigr)

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

\begin{align}\log\prod_{j=m+1}^k\Bigl(1+\frac{r-1}j\Bigr) &\le\sum_{j=m+1}^k\frac{r-1}j\\ &\le(r-1)\int_{m+1}^{k+1}\frac{dx}x =\log\Bigl(\frac{k+1}{m+1}\Bigr)^{r-1}. \end{align}

Therefore,

\sum_{k=m}^\infty\biggl|\binom{k+r-1}k\biggr| \le\biggl|\prod_{j=1}^m\frac{k+r-1}j\biggr| \sum_{k=m}^\infty\Bigl(\frac{m+1}{k+1}\Bigr)^{1-r}<\infty

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}^\infty\binom{-r}k(-x)^k

holds for all x in [–1,1]. In particular, using the abbreviation q = 1 − p, 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-qs)^{-r}-\sum_{j=0}^{m-1}\binom{j+r-1}j (qs)^j} {p^{-r}-\sum_{j=0}^{m-1}\binom{j+r-1}j q^j} \qquad\text{for } |s|\le\frac1q.\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^{-r}} \qquad\text{for }|s|\le\frac1q.

[edit] References

  1. ^ Hess, Klaus Th.; Anett Liewald, Klaus D. Schmidt (2002). "An extension of Panjer's recursion" (PDF). ASTIN Bulletin 32 (2): 283–297. 
  2. ^ Willmot, Gordon (1988). "Sundt and Jewell's family of discrete distributions" (PDF). ASTIN Bulletin 18 (1): 17–29. 
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Probability distributions
 
Discrete univariate with infinite support