The Panjer recursion is an algorithm to compute the probability distribution of a compound random variable
where both and 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 of Harry Panjer[1]. It is heavily used in actuarial science.
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We are interested in the compound random variable where and fulfill the following preconditions.
We assume the to be i.i.d. and independent of . Furthermore the have to be distributed on a lattice with latticewidth .
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 a and b which fulfill . The initial value is determined such that
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.
The algorithm now gives a recursion to compute the .
The starting value is with the special cases
and
and proceed with
The following example shows the approximated density of where and with lattice width h = 0.04. (See Fréchet distribution.)