(a,b,0) class of distributions

In probability theory, the distribution of a discrete random variable N whose values are nonnegative integers is said to be a member of the (a, b, 0) class of distributions if its probability mass function obeys

${\displaystyle {\frac {p_{k}}{p_{k-1}}}=a+{\frac {b}{k}},\qquad k=1,2,3,\dots }$

where ${\displaystyle p_{k}=P(N=k)}$ (provided ${\displaystyle a}$ and ${\displaystyle b}$ exist and are real).

There are only three discrete distributions that satisfy the full form of this relationship: the Poisson, binomial and negative binomial distributions. These are also the three discrete distributions among the six members of the natural exponential family with quadratic variance functions (NEF–QVF).

More general distributions can be defined by fixing some initial values of p_j and applying the recursion to define subsequent values. This can be of use in fitting distributions to empirical data. However, some further well-known distributions are available if the recursion above need only hold for a restricted range of values of k:[1] for example the logarithmic distribution and the discrete uniform distribution.

The (a, b, 0) class of distributions has important applications in actuarial science in the context of loss models.[2]

Properties

Sundt[3] proved that only the binomial distribution, the Poisson distribution and the negative binomial distribution belong to this class of distributions, with each distribution being represented by a different sign of a. Furthermore, it was shown by Fackler[4] that there is a universal formula for all three distributions, called the (united) Panjer distribution.

The more usual parameters of these distributions are determined by both a and b. The properties of these distributions in relation to the present class of distributions are summarised in the following table. Note that ${\displaystyle W_{N}(x)\,}$ denotes the probability generating function.

Distribution	${\displaystyle P[N=k]\,}$	${\displaystyle a\,}$	${\displaystyle b\,}$	${\displaystyle p_{0}\,}$	${\displaystyle W_{N}(x)\,}$	${\displaystyle E[N]\,}$	${\displaystyle Var(N)\,}$
Binomial	${\displaystyle {\binom {n}{k}}p^{k}(1-p)^{n-k}}$	${\displaystyle {\frac {-p}{1-p}}}$	${\displaystyle {\frac {p(n+1)}{1-p}}}$	${\displaystyle (1-p)^{n}\,}$	${\displaystyle (px+(1-p))^{n}\,}$	${\displaystyle np\,}$	${\displaystyle np(1-p)\,}$
Poisson	${\displaystyle e^{-\lambda }{\frac {\lambda ^{k}}{k!}}\,}$	${\displaystyle 0\,}$	${\displaystyle \lambda \,}$	${\displaystyle e^{-\lambda }\,}$	${\displaystyle e^{\lambda (x-1)}\,}$	${\displaystyle \lambda \,}$	${\displaystyle \lambda \,}$
Negative binomial	${\displaystyle {\frac {\Gamma (r+k)}{k!\,\Gamma (r)}}\,p^{r}\,(1-p)^{k}\,}$	${\displaystyle 1-p\,}$	${\displaystyle (1-p)(r-1)\,}$	${\displaystyle p^{r}\,}$	${\displaystyle \left({\frac {p}{1-x(1-p)}}\right)^{r}\,}$	${\displaystyle {\frac {r(1-p)}{p}}\,}$	${\displaystyle {\frac {r(1-p)}{p^{2}}}\,}$
Panjer distribution	${\displaystyle \left(1+{\frac {\lambda }{\alpha }}\right)^{-\alpha }{\frac {\lambda ^{k}}{k!}}\prod _{i=0}^{k-1}{\frac {\alpha +i}{\alpha +\lambda }}\,}$	${\displaystyle {\frac {\lambda }{\alpha +\lambda }}\,}$	${\displaystyle {\frac {(\alpha -1)\lambda }{\alpha +\lambda }}\,}$	${\displaystyle \left(1+{\frac {\lambda }{\alpha }}\right)^{-\alpha }\,}$	${\displaystyle \left(1-{\frac {\lambda }{\alpha }}(x-1)\right)^{-\alpha }\,}$	${\displaystyle \lambda \,}$	${\displaystyle \lambda \left(1+{\frac {\lambda }{\alpha }}\right)\,}$

Note that the Panjer distribution reduces to the Poisson distribution in the limit case ${\displaystyle \alpha \rightarrow \pm \infty }$; it coincides with the negative binomial distribution for positive, finite real numbers ${\displaystyle \alpha \in \mathbb {R} _{>0}}$, and it equals the binomial distribution for negative integers ${\displaystyle -\alpha \in \mathbb {Z} }$.

Plotting

An easy way to quickly determine whether a given sample was taken from a distribution from the (a,b,0) class is by graphing the ratio of two consecutive observed data (multiplied by a constant) against the x-axis.

By multiplying both sides of the recursive formula by ${\displaystyle k}$, you get

${\displaystyle k\,{\frac {p_{k}}{p_{k-1}}}=ak+b,}$

which shows that the left side is obviously a linear function of ${\displaystyle k}$. When using a sample of ${\displaystyle n}$ data, an approximation of the ${\displaystyle p_{k}}$'s need to be done. If ${\displaystyle n_{k}}$ represents the number of observations having the value ${\displaystyle k}$, then ${\displaystyle {\hat {p}}_{k}={\frac {n_{k}}{n}}}$ is an unbiased estimator of the true ${\displaystyle p_{k}}$.

Therefore, if a linear trend is seen, then it can be assumed that the data is taken from an (a,b,0) distribution. Moreover, the slope of the function would be the parameter ${\displaystyle a}$, while the ordinate at the origin would be ${\displaystyle b}$.

See also

• Panjer recursion

References

1. Hess, Klaus Th.; Liewald, Anett; Schmidt, Klaus D. (2002). "An extension of Panjer's recursion" (PDF). ASTIN Bulletin. 32 (2): 283–297. doi:10.2143/AST.32.2.1030. Archived (PDF) from the original on 2009-06-20. Retrieved 2009-06-18.
2. Klugman, Stuart; Panjer, Harry; Gordon, Willmot (2004). Loss Models: From Data to Decisions. Series in Probability and Statistics (2nd ed.). New Jersey: Wiley. ISBN 978-0-471-21577-6.
3. Sundt, Bjørn; Jewell, William S. (1981). "Further results on recursive evaluation of compound distributions" (PDF). ASTIN Bulletin. 12 (1): 27–39. doi:10.1017/S0515036100006802.
4. Fackler, Michael (2009). "Panjer class united - one formula for the Poisson, Binomial and Negative Binomial distribution" (PDF). ASTIN Colloquium. International Actuarial Association.