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 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

For a natural number $m \geq 1$ and real parameters $p$ , $r$ with $0 < p \leq 1$ and $- m < r < - m + 1$ , the probability mass function of the 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} 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 $Γ$ denotes the gamma function.

Probability generating function

Using that $f (.; m, r, ps)$ for $s \in$ $(0, 1]$ 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 $m = 1$ , hence $r \in$ $(-1, 0)$ , this simplifies to

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

References

↑ 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.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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