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 foundation of the Panjer recursion for computing aggregate loss distributions. The three distributions are uniquely identified by their (a, b) values.

Poisson with parameter lambda: a = 0, b = lambda. Since a = 0, the ratio p_k/p_{k-1} = lambda/k, which matches the well-known Poisson probability ratio. Negative binomial with parameters r > 0 and beta > 0: a = beta/(1+beta) > 0, b = (r-1)*beta/(1+beta). Binomial with parameters n and q: a = -q/(1-q) < 0, b = (n+1)*q/(1-q).

The sign of a determines which distribution: a = 0 is Poisson, a > 0 is negative binomial, a < 0 is binomial. This classification is worth memorizing because it immediately identifies which frequency distribution is compatible with a given recursive relationship.

The Poisson Distribution

The Poisson distribution with parameter lambda has PMF p_k = e^(-lambda) * lambda^k / k! for k = 0, 1, 2, ... Its mean and variance are both equal to lambda. The Poisson is appropriate when events occur independently at a constant rate, which is a reasonable starting model for many insurance claim processes.

The Poisson has several useful properties. The sum of independent Poisson random variables is Poisson: if N1 ~ Poisson(lambda_1) and N2 ~ Poisson(lambda_2) independently, then N1 + N2 ~ Poisson(lambda_1 + lambda_2). Thinning: if N ~ Poisson(lambda) and each event is independently retained with probability p, the number of retained events is Poisson(lambda*p).

The main limitation of the Poisson for insurance applications is the mean-equals-variance property. Real insurance data almost always shows more variability in claim counts than the Poisson predicts, a phenomenon called overdispersion.

The Negative Binomial Distribution

The negative binomial with parameters r > 0 and beta > 0 has mean r*beta and variance r*beta*(1+beta). Since variance = mean*(1+beta) > mean, the negative binomial always exhibits overdispersion. The degree of overdispersion increases with beta: when beta is large, the variance is much larger than the mean.

A key result for Exam STAM is that a Poisson-Gamma mixture yields a negative binomial. If N|Lambda ~ Poisson(Lambda) and Lambda ~ Gamma(r, beta), then the unconditional distribution of N is negative binomial with parameters r and beta. This provides a compelling justification for using the negative binomial: claim counts follow a Poisson process for each individual, but the Poisson rate varies across the population according to a Gamma distribution. The resulting heterogeneity produces overdispersion.

The geometric distribution is the special case r = 1 of the negative binomial. It has PMF p_k = (1/(1+beta)) * (beta/(1+beta))^k and is the discrete analogue of the exponential distribution (memoryless property for counting).

The Binomial Distribution

The binomial with parameters n (positive integer) and q (probability, 0 < q < 1) has mean nq and variance nq(1-q). Since variance = mean*(1-q) < mean, the binomial exhibits underdispersion. The binomial applies when there is a fixed maximum number of trials and each trial independently results in a claim with probability q.

In insurance, the binomial is appropriate for portfolios where each policy can produce at most one claim in the observation period. With n policies and individual claim probability q, the total number of claims follows a Binomial(n, q) distribution. This is the basis of the individual risk model.

The binomial is less commonly used than the Poisson or negative binomial for aggregate claim frequency because insurance data rarely exhibits underdispersion. However, it appears in credibility calculations and in the individual risk model for computing aggregate loss distributions.

Extended Distributions: Zero-Modification

The (a, b, 1) class extends the (a, b, 0) class by allowing p_0 to be arbitrary rather than determined by the recursion. This produces zero-modified distributions. If p_0 is larger than the unmodified value, the distribution is zero-inflated (excess zeros). If p_0 = 0, the distribution is zero-truncated (no zeros possible).

Zero-inflated models are common in insurance because many policyholders never file a claim. A zero-inflated Poisson (ZIP) model assumes that with probability phi, the count is zero (representing "safe" risks who never claim), and with probability 1 - phi, the count follows a Poisson (representing "active" risks). The result has P(N=0) = phi + (1-phi)*e^(-lambda), which exceeds the Poisson P(N=0) = e^(-lambda).

Zero-truncated distributions arise when the data collection process only records positive counts (e.g., data from claim files, where every record has at least one claim). The Panjer recursion can be adapted for (a, b, 1) class distributions, with a modified initial condition.

Parameter Estimation and Goodness of Fit

For the Poisson, the MLE of lambda is the sample mean. For the negative binomial with parameters r and beta, method of moments estimates use the sample mean and variance: set r*beta = x_bar and r*beta*(1+beta) = s^2. Solving gives beta_hat = s^2/x_bar - 1 and r_hat = x_bar/beta_hat. MLEs require numerical optimization of the log-likelihood.

The chi-squared goodness-of-fit test compares observed frequencies to expected frequencies: chi^2 = sum of (O_k - E_k)^2 / E_k, where the sum is over all cells (grouping cells with small expected counts). The test statistic follows a chi-squared distribution with degrees of freedom equal to (number of cells - 1 - number of estimated parameters). Rejecting H0 suggests the fitted distribution does not adequately describe the data.

Study Tips and Common Pitfalls

The most common mistake is confusing the parameterization of the negative binomial. Different textbooks use different conventions. On Exam STAM, the standard parameterization uses r and beta, with mean = r*beta and variance = r*beta*(1+beta). Be certain you are using the correct parameterization for the formulas on the exam.

Another frequent error is applying the Panjer recursion with incorrect (a, b) values. Always verify which distribution you are working with and compute a and b accordingly before starting the recursion. Practice at least 5 problems fitting frequency distributions to data and 3 problems applying the Panjer recursion. These computational skills are heavily tested on Exam STAM.