Panjer recursion: Difference between revisions

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Revision as of 18:47, 24 March 2017

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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:

:<math>P[N=k] = p_k= \left(a + \frac{b}{k} \right) \cdot p_{k-1},~~k \ge 1.\, </math>

for some ''<math>a''</math> and ''<math>b''</math> which fulfill <math>a+b \ge 0\,</math>. The initial value <math>p_0\,</math> is determined such that <math>\sum_{k=0}^\infty p_k = 1.\,</math>

The Panjer recursion makes use of this iterative relationship to specify a recursive way of constructing the probability distribution of ''S''. In the following <math>W_N(x)\,</math> denotes the [[probability generating function]] of ''N'': for this see the table in [[(a,b,0) class of distributions]].