Deriving the Gamma Severity Likelihood

From a Severity Model to a Likelihood

A severity model specifies a distribution for claim amounts, conditional on the characteristics used to rate or classify the claims. Its likelihood answers a related question: how well do particular model parameters account for the observed severities? Writing the likelihood carefully matters because a Gamma model is parameterized by both a mean and a dispersion, and its scale is not generally equal to its mean.

Here, each observation is one positive claim severity. The likelihood is conditional on the claims’ recorded rating variables, and the model treats the severities as conditionally independent. This is the ordinary claim-level Gamma likelihood: it does not by itself adjust for claims selected by a deductible, censored at a policy limit, or otherwise observed through a selection process.

The shape-scale convention means that a Gamma variable with shape \(\alpha\) and scale \(\theta\) has mean \(\alpha\theta\). In this model, \((1/\phi)(\phi\mu_j)=\mu_j\), as required, while its variance is \((1/\phi)(\phi\mu_j)^2=\phi\mu_j^2\). Thus dispersion changes the spread around the mean without changing the specified mean.

The Individual Density Contribution

For \(y_j>0\), the Gamma density contribution of observation \(j\), conditional on its rating variables, is

Here \(\Gamma(\cdot)\) is the gamma function, with \(\Gamma(k)=(k-1)!\) for positive integers \(k\). The density is evaluated at the observed severity; it is not the probability that a continuous claim amount equals that exact value. A density can exceed one, and its units are the inverse of the units used for severity.

If there are \(n\) conditionally independent claim severities \(y_1,\ldots,y_n\), each with its own fitted mean \(\mu_j\) but a common dispersion \(\phi\), the joint likelihood is the product of the individual density contributions:

The symbol \(L\) denotes a function of the model parameters when the observed \(y_j\)'s are held fixed. In a severity GLM, the means are not arbitrary: for a specified link \(g\) and coefficient vector \(\boldsymbol{\beta}\), \(\mu_j=g^{-1}(\mathbf{x}_j^\mathsf{T}\boldsymbol{\beta})\). Substituting these means and a value of \(\phi\) into the likelihood gives the likelihood for that GLM parameter setting.

Deriving the Log-Likelihood

Products of many densities can be cumbersome to calculate. Taking a natural logarithm turns the product into a sum. Applying the logarithm to each factor gives

This expression shows where the data and parameters enter. Each observation contributes a term involving its observed severity \(y_j\), its fitted mean \(\mu_j\), and the dispersion \(\phi\). The gamma-function term and the powers involving \(\phi\) are part of the density’s normalizing constant; they must be retained when comparing different dispersion values.

Why the log-likelihood has this form. Conditional independence means the joint density factors into the product of the individual densities. Taking its logarithm gives the sum of their logarithms because \(\log(ab)=\log a+\log b\). For one factor, the logarithm of the numerator power is \((1/\phi-1)\log y_j\); the exponential contributes \(-y_j/(\phi\mu_j)\); and the denominator contributes \(-\log\Gamma(1/\phi)-(1/\phi)\log(\phi\mu_j)\). Adding these terms for all claims yields the displayed expression. This derivation also explains why dropping a normalizing term can change likelihood comparisons, particularly when comparing dispersions.

Worked Likelihood Calculations

Assumptions and Exam Traps

The ordinary likelihood above requires positive continuous severities modeled as Gamma and observations that are conditionally independent given their rating variables. It also assumes the observed claims are the severity observations described by the model. If the data have been selected by a deductible or are censored at a limit, the ordinary density at the recorded amount need not be the correct likelihood contribution; the observation process must be represented as well.

A frequent exam error is to treat \(\phi\) as the scale, or to substitute the mean for the scale. The scale is \(\phi\mu_j\), while the shape is \(1/\phi\). For example, when \(\phi=0.5\) and \(\mu_j=4\), the shape is \(2\) and the scale is \(2\), not \(0.5\) or \(4\). That distinction determines both the exponential term and the normalizing power in the density.

Another trap is to omit terms because they look constant. When estimating regression coefficients with a fixed dispersion, a term independent of those coefficients may sometimes be excluded from a particular optimization calculation. But when writing the full likelihood, or comparing different dispersion values, retain the complete expression unless the comparison justifies removing a term common to every candidate.