Panjer distribution

Panjer distribution
Distribution function
parameter	from
carrier	${\displaystyle \mathbb {N} \cup \{0\}}$
Expected value	${\displaystyle {\frac {a+b}{1-a}}}$
Variance	${\displaystyle {\frac {a+b}{(1-a)^{2}}}}$

The Panjer distribution (after Harry Panjer ) is a probability distribution that combines the distributions of negative binomial distribution , binomial distribution and Poisson distribution in one distribution class . Thus it belongs to the univariate discrete probability distributions . It is used in actuarial mathematics as a loss number distribution, since its special recursive structure enables an efficient algorithm to calculate the total loss distribution of an insurance portfolio.

characterization

The class of the Panjer distribution consists of all distributions for which there are constants with , so that the following recursion rule applies to the counting density : ${\displaystyle \mathbb {N} _{0}}$ ${\displaystyle a,b\in \mathbb {R} }$ ${\displaystyle a+b\geq 0}$ ${\displaystyle p_{k}=P(X=k)}$

${\displaystyle p_{k}=\left(a+{\frac {b}{k}}\right)\cdot p_{k-1},~~k\geq 1.}$

The probability results from the normalization condition ${\displaystyle p_{0}}$

${\displaystyle \sum _{k=0}^{\infty }p_{k}=1.}$

properties

The expectation and variance of the Panjer distribution are given by

${\displaystyle E(X)={\frac {a+b}{1-a}},~~V(X)={\frac {a+b}{(1-a)^{2}}}.}$

It is

${\displaystyle {\frac {V(X)}{E(X)}}={\frac {1}{1-a}},}$

from which it follows that

${\displaystyle V(X)>E(X)~~\iff a>0.}$

${\displaystyle V(X)=E(X)~~\iff a=0.}$

${\displaystyle V(X)<E(X)~~\iff a<0.}$

Special cases

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

With you get the Poisson distribution . So in this case . ${\displaystyle a=0,~b=\lambda ,~p_{0}=e^{-\lambda }}$ ${\displaystyle V(X)=E(X)}$

Panjer and binomial distribution

With you get the binomial distribution . In this case it is . ${\displaystyle a=-{\frac {p}{1-p}},~b=(n+1)\cdot {\frac {p}{1-p}},~p_{0}=(1-p)^{n}}$ ${\displaystyle V(X)<E(X)}$

With you get the negative binomial distribution (counting the failures). Here is now . ${\displaystyle a=1-p,~b=(r-1)\cdot (1-p),~p_{0}=p^{r}}$ ${\displaystyle V(X)>E(X)}$

See also

• Panjer algorithm

literature

• Thomas Mack: Property Insurance Mathematics . 2nd edition, Verlag Versicherungswirtschaft 2002, ISBN 3-88487-957-X .