Panjer recursion

We are interested in the compound random variable ${\displaystyle S=\sum _{i=1}^{N}X_{i}\,}$ where ${\displaystyle N\,}$ and ${\displaystyle X_{i}\,}$ fulfill the following preconditions.

The algorithm now gives a recursion to compute the ${\displaystyle g_{k}=P[S=hk]\,}$.

The starting value is ${\displaystyle g_{0}=W_{N}(f_{0})\,}$ with the special cases

${\displaystyle g_{0}=p_{0}\cdot \exp(f_{0}b)\quad {\text{ if }}\quad a=0,\,}$

and

${\displaystyle g_{0}={\frac {p_{0}}{(1-f_{0}a)^{1+b/a}}}\quad {\text{ for }}\quad a\neq 0,\,}$

and proceed with

${\displaystyle g_{k}={\frac {1}{1-f_{0}a}}\sum _{j=1}^{k}\left(a+{\frac {b\cdot j}{k}}\right)\cdot f_{j}\cdot g_{k-j}.\,}$

The following example shows the approximated density of ${\displaystyle \scriptstyle S\,=\,\sum _{i=1}^{N}X_{i}}$ where ${\displaystyle \scriptstyle N\,\sim \,{\text{NegBin}}(3.5,0.3)\,}$ and ${\displaystyle \scriptstyle X\,\sim \,{\text{Frechet}}(1.7,1)}$ with lattice width h = 0.04. (See Fréchet distribution.)

As observed, an issue may arise at the initialization of the recursion. Guégan and Hassani (2009) have proposed a solution to deal with that issue .[5]

• Panjer recursion and the distributions it can be used with