Panjer recursion

Panjer the recursion (or Panjer ) algorithm is an algorithm of the probability distribution of a special composite random variable

To calculate. In this case, and are random variables which make up a collective model and designates the indicator function.

The algorithm was first published in a publication by Harry Panjer. It is often used in insurance.

Preconditions

We are interested in the special composite random variable must satisfy where and are the following preconditions:

Number of claims distribution

Is a "damage number distribution ", ie. is independent of.

Furthermore, an element of the Panjer class must be. The Panjer class consists of all random variables with values ​​in that satisfy the following relation: with and for and with. The value is determined such that is satisfied.

Sundt proved in the paper that only the binomial distribution, the Poisson distribution and the negative binomial distribution are the Panjer class. You have described the parameters and values ​​in the following table, the probability generating function called.

Individual loss distribution

We assume that independent and identically distributed random variables, which are independent of. Furthermore, it must be distributed on a grid with grid length.

Recursion

The algorithm uses a recursion in order to calculate the probabilities.

The starting value is:

The following values ​​can be calculated as follows:

Example

The example shows the approximate density function of where and. The individual loss distribution was discretized with a grid width ( see also Fréchet distribution ).