Frequency Distributions: Poisson, Negative Binomial, Binomial for STAM

The (a, b, 0) Class

The Poisson, negative binomial, and binomial distributions belong to the (a, b, 0) class, meaning their probability ratios satisfy p_k/p_{k-1} = a + b/k for k = 1, 2, 3, ... This recursive relationship is the basis of the Panjer recursion for computing aggregate loss distributions. Poisson: a = 0, b = lambda. Negative Binomial: a = beta/(1+beta), b = (r-1)*beta/(1+beta). Binomial: a = -q/(1-q), b = (n+1)*q/(1-q), where q is the success probability.

Comparing the Three Distributions

The Poisson has mean = variance = lambda and is appropriate when occurrences are independent at a constant rate. The negative binomial has variance > mean (overdispersion), making it suitable when claim counts show more variability than the Poisson predicts. This often arises from heterogeneous risk populations. The binomial has variance < mean (underdispersion) and applies when there is a maximum possible count (e.g., n policies each filing at most one claim).

In practice, the negative binomial is used more often than the binomial because insurance data typically exhibits overdispersion. A Poisson-Gamma mixture yields a negative binomial, providing a theoretical justification.

Parameter Estimation

For the Poisson, the MLE of lambda is the sample mean. For the negative binomial with parameters r and beta, moment estimates use the sample mean and variance: mean = r*beta and variance = r*beta*(1+beta). Solve for r and beta. MLEs require numerical optimization. For Exam STAM, be comfortable fitting these distributions to data and performing goodness-of-fit tests (chi-squared test) to assess whether the chosen distribution adequately describes the observed claim counts.