The Panjer recursion is an algorithm to compute the probability distribution approximation of a compound random variable
are random variables and of special types.
In more general cases the distribution of S is a compound distribution.
The recursion for the special cases considered was introduced in a paper [1] by Harry Panjer (Distinguished Emeritus Professor, University of Waterloo[2]).
It is heavily used in actuarial science (see also systemic risk).
We are interested in the compound random variable
fulfill the following preconditions.
have to be distributed on a lattice
In actuarial practice,
is obtained by discretisation of the claim density function (upper, lower...).
The number of claims N is a random variable, which is said to have a "claim number distribution", and which can take values 0, 1, 2, .... etc.. For the "Panjer recursion", the probability distribution of N has to be a member of the Panjer class, otherwise known as the (a,b,0) class of distributions.
This class consists of all counting random variables which fulfill the following relation: for some
The initial value
The Panjer recursion makes use of this iterative relationship to specify a recursive way of constructing the probability distribution of S. In the following
denotes the probability generating function of N: for this see the table in (a,b,0) class of distributions.
In the case of claim number is known, please note the De Pril algorithm.
[3] This algorithm is suitable to compute the sum distribution of
discrete random variables.
[4] The algorithm now gives a recursion to compute the
with the special cases and and proceed with The following example shows the approximated density of
(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 .