For positive integer severities with probability mass function and independent count in the Panjer claim-count class, the aggregate probability mass function starts at and satisfies the displayed recurrence. If are the count, severity and aggregate probability generating functions, respectively, then and . Equating the coefficient of gives . The positivity assumption excludes zero severities; otherwise the initial value and recursion denominator change.
For a claim count distribution in the Panjer claim-count class and independent nonnegative integer severities with mass , the probability generating function composition is . Combining it with gives . Equating coefficients yields the displayed recursion. Zero severities change both the initial mass and the denominator. For nondegenerate proper Panjer count laws , so the denominator is positive. An identically zero count is handled directly if its redundant parameter choice makes the denominator vanish.
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Panjer recursion is a recursive algorithm used in actuarial science and insurance mathematics to calculate the distribution of the sum of independent random variables, particularly in the context of risk management and insurance claims. Named after Hendrik Panjer, this method is particularly useful for computing the probabilities associated with different outcomes of aggregate claims. ### Key Elements of Panjer Recursion: 1. **Assumptions**: - The random variables (e.g., claims) are independent.