Worked Complete-Data Frequency Estimation

Introduction

Claim-frequency estimation starts with a clear statement of what one observation represents and how much exposure it contributes. A count of three claims over one policy-year is not equivalent to a count of three claims over ten policy-years. The likelihood must reflect both the observed counts and the model’s assumptions about exposure.

This tutorial develops maximum likelihood estimates for three standard frequency models: binomial, Poisson, and negative binomial. Each worked problem uses complete count data: every count is observed, and the relevant trial totals or exposures are known. The focus is on constructing the likelihood, solving its score equation, and checking that the result is a maximum.

Three Count Models and Their Likelihoods

For a binomial frequency model, each observation has a fixed number \(m\) of opportunities for a claim, and each opportunity produces at most one claim. The model assumes independent opportunities with the same claim probability \(p\). Thus \(K_i\sim\operatorname{Binomial}(m,p)\), where \(K_i\) is the claim count for observation \(i\). For \(N\) observations, write \(K=\sum_{i=1}^N K_i\). The likelihood, log-likelihood, and score are

Here \(C\) does not depend on \(p\). Setting the score to zero gives \(\widehat p=K/(Nm)\), provided \(0<K<Nm\). The estimate is the observed number of claim-producing opportunities divided by the total number of opportunities.

For a Poisson model, observation \(i\) has known exposure \(e_i>0\), and its count satisfies \(K_i\sim\operatorname{Poisson}(\lambda e_i)\). The rate \(\lambda\) is claims per exposure unit. This model assumes independent counts and a common rate per unit of exposure. With \(K=\sum_i K_i\) and \(E=\sum_i e_i\), the log-likelihood and score are

For \(K>0\), the score equation gives \(\widehat\lambda=K/E\). It is total claims divided by total exposure, not the unweighted average of the individual rates.

A negative binomial model can represent counts with greater variability than a Poisson model. In this tutorial, \(K_i\) is the number of failures before \(r\) successes, where \(r>0\) is known and common to all observations. Define \(p\) as the probability of a success. Then

For \(N\) independent counts and \(K=\sum_i K_i\), the log-likelihood, apart from terms not involving \(p\), and its score are

When \(K>0\), solving \(U(p)=0\) yields \(\widehat p=Nr/(Nr+K)\). The fitted mean count per observation is then \(r(1-\widehat p)/\widehat p\).

Why the Score Equations Reduce to Totals

The individual binomial likelihood contributions contain combinatorial coefficients, and the individual Poisson and negative binomial contributions contain terms involving the separate counts. But these terms do not all depend on the parameter. For example, in the Poisson model, multiplying the probabilities gives a factor \(\prod_i e_i^{K_i}/K_i!\), which is constant with respect to \(\lambda\), and the parameter-dependent part is \(\lambda^{\sum_i K_i}\exp(-\lambda\sum_i e_i)\). Taking logarithms therefore produces \(K\log\lambda-\lambda E\), so differentiating gives \(K/\lambda-E\).

The same principle explains the other score equations: after collecting parameter-dependent factors, the data enter through the total count \(K\) and the total number of binomial opportunities \(Nm\) or total exposure \(E\). This is why the score equations involve these totals, rather than a separate parameter contribution for each observation.

A score equation identifies a stationary point, not automatically a maximum. For the interior estimates here, the second derivatives are

Each is negative throughout the interior when its relevant counts are positive, so the log-likelihood is strictly concave there. This confirms that an interior solution is a maximum, rather than a minimum or another stationary point.

Worked Frequency Estimation

Boundary Cases and Exam Traps

The score equation is an interior condition. If the data are at an endpoint, the parameter space must be checked directly. For the binomial model, if \(K=0\), the likelihood is proportional to \((1-p)^{Nm}\); if \(K=Nm\), it is proportional to \(p^{Nm}\). On the closed parameter space \(0\leq p\leq1\), the maxima occur at \(p=0\) and \(p=1\), respectively. If the parameter space is restricted to \(0<p<1\), these endpoint cases have suprema approached at the boundary but no attained maximum in that open space.

For a Poisson model with \(K=0\), the likelihood is proportional to \(\exp(-\lambda E)\), which decreases as \(\lambda\) increases. If \(\lambda=0\) is allowed, the maximum is at \(\lambda=0\); if only \(\lambda>0\) is allowed, the likelihood has a supremum as \(\lambda\) approaches zero but no maximum in that open parameter space.

The negative binomial boundary case requires similar care. When \(K=0\), its likelihood is proportional to \(p^{Nr}\), which increases as \(p\) approaches 1. On the closed parameter space \(0<p\leq1\), the maximum occurs at \(p=1\), corresponding to a degenerate count of zero. Under the stated open parameter space \(0<p<1\), the likelihood has a supremum as \(p\to1\), but no maximum.

A second frequent trap is to overlook the model’s exposure structure. A binomial count has a fixed number of claim opportunities; a Poisson count has a rate multiplied by known exposure; the negative binomial model here uses a known shape \(r\) and a specified success-probability convention. Applying the right algebra to the wrong model still gives the wrong estimate.