Choosing Between Gamma and Inverse Gaussian Severity Models

Why Compare Severity Variance Patterns?

Two severity models can describe the same positive conditional mean while implying different amounts of variation around that mean. That distinction matters in ratemaking: a model that fits average claim size well may still misrepresent the spread of claim sizes across risk groups. For choosing between the Gamma and inverse Gaussian models, the key comparison is how conditional variance changes as conditional mean changes.

Both models can use a log link to ensure that fitted mean severities are positive. The link specifies how the mean depends on rating variables; it does not determine which variance pattern is appropriate. A useful comparison therefore asks whether observed variation is approximately proportional to the square or the cube of the conditional mean.

Definition: Let $Y$ be a positive claim severity and let $x$ denote its rating variables. Write $\mu(x)=E[Y\mid x]$. Under the Gamma variance convention, $\operatorname{Var}(Y\mid x)=\phi\mu(x)^2$, where $\phi>0$ is the dispersion parameter. Under the inverse Gaussian convention, $\operatorname{Var}(Y\mid x)=\phi\mu(x)^3$, where $\phi>0$. Equivalently, the inverse Gaussian shape parameter is $\lambda=1/\phi$, giving variance $\mu(x)^3/\lambda$.

These are conditional statements: they describe variation among claims with the same modeled rating characteristics, not variation caused by mixing together groups with different means. The Gamma model has a constant conditional coefficient of variation, while the inverse Gaussian model’s coefficient of variation increases with the mean:

\operatorname{CV}_{\mathrm{Gamma}}=\frac{\sqrt{\operatorname{Var}(Y\mid x)}}{\mu(x)}=\sqrt{\phi}, \qquad \operatorname{CV}_{\mathrm{IG}}=\sqrt{\phi\mu(x)}.

Thus, the Gamma model says that relative variability is stable across mean levels. The inverse Gaussian model says that relative variability grows as mean severity grows. These are patterns to compare with conditional data; neither pattern is automatically preferable for insurance severity.

How to Use Grouped Mean-Variance Evidence

Suppose claims are divided into groups that are reasonably homogeneous with respect to the rating variables, and each group has enough claims to estimate its conditional mean and variance. For group $j$, let the estimates be $\widehat{\mu}_j$ and $s_j^2$. The diagnostic quantities are:

\widehat{\phi}_{\mathrm{Gamma},j}=\frac{s_j^2}{\widehat{\mu}_j^2}, \qquad \widehat{\phi}_{\mathrm{IG},j}=\frac{s_j^2}{\widehat{\mu}_j^3}.

If the Gamma pattern is plausible, the first ratio should be reasonably stable across groups. If the inverse Gaussian pattern is plausible, the second ratio should be reasonably stable. “Reasonably stable” is important: sample variances are noisy, especially for small groups, and these ratios are diagnostics rather than exact tests.

The same comparison can be viewed on a log scale. If variance is approximately proportional to $\mu^p$, the slope in a plot of log conditional variance against log conditional mean is approximately $p$. Gamma corresponds to $p=2$, and inverse Gaussian corresponds to $p=3$. This visual summary is useful, but it does not replace checking that the groups represent comparable conditional populations.

Worked Example: Stable Relative Variation Supports Gamma

A hypothetical insurer examines three groups of 100 independent, uncensored claim severities each. Within each group, the claims are treated as having the same rating characteristics. The observed group means are $100, $200, and $400, and the corresponding sample variances are $400, $1,600, and $6,400. Assume the target is the same uncensored severity measure in all three groups. Compare the two variance patterns.

Method. Calculate each model’s dispersion ratio by dividing the observed variance by the relevant power of the observed mean.

For the Gamma pattern, the three ratios are:

\frac{400}{100^2}=0.04,\qquad \frac{1{,}600}{200^2}=0.04,\qquad \frac{6{,}400}{400^2}=0.04.

The ratios are constant. A Gamma model with $\phi=0.04$ predicts variances $0.04(100^2)=400$, $0.04(200^2)=1{,}600$, and $0.04(400^2)=6{,}400$. In the shape-scale parameterization, $\alpha=1/\phi=25$ and the group-specific scale is $\theta=\phi\mu$, or $4, $8, and $16, respectively. Each resulting Gamma distribution has the stated group mean and variance.

For the inverse Gaussian pattern, the ratios are:

\frac{400}{100^3}=0.0004,\qquad \frac{1{,}600}{200^3}=0.0002,\qquad \frac{6{,}400}{400^3}=0.0001.

These are not stable. The Gamma variance pattern is better supported by these summaries. The conclusion is not that every claim follows a Gamma distribution; it is that, among these two candidate variance functions, the observed conditional variation follows the square-of-the-mean pattern more closely.

Check. The Gamma predictions reproduce all three reported variances, and the inverse Gaussian ratios decrease by a factor of two each time the mean doubles. The common mistake is to compare raw variances alone: the variances increase in both candidate models, so their increase by itself does not distinguish the models.

Why the Ratios Work

The ratios above follow directly by rearranging each model’s conditional variance equation. This is the key justification for using them: if the model’s variance equation is correct and its dispersion is constant across the groups, dividing the variance by its specified mean function should remove the effect of mean level.

Start with the Gamma assumption.
If $\operatorname{Var}(Y\mid x)=\phi\mu(x)^2$, divide both sides by $\mu(x)^2$. Then $\operatorname{Var}(Y\mid x)/\mu(x)^2=\phi$, which does not depend on $x$.

Start with the inverse Gaussian assumption.
If $\operatorname{Var}(Y\mid x)=\phi\mu(x)^3$, divide both sides by $\mu(x)^3$. Then $\operatorname{Var}(Y\mid x)/\mu(x)^3=\phi$, also constant across $x$ under that model.

Replace unknown conditional moments with estimates.
Within sufficiently large, comparable groups, use the sample mean and sample variance as estimates of the conditional moments. Sampling noise means the resulting ratios will not generally be identical, even when a variance pattern is appropriate.

The derivation depends on comparing conditional variation at different means. If groups differ in unmodeled risk characteristics, or if their severities are selected differently, their sample variances may not estimate comparable conditional variances. In particular, severity data subject to deductibles, limits, truncation, censoring, or sampling rules must be defined and handled consistently with the target being modeled. Otherwise, a changing selection mechanism can look like a change in the variance function.

Worked Example: Increasing Relative Variation Supports Inverse Gaussian

A hypothetical portfolio has three homogeneous groups, each containing 120 independent positive severities observed on the same uncensored basis. Their sample means are $100, $200, and $400; their sample variances are $1,000, $8,000, and $64,000. Assume the conditional means are adequately represented and compare the two candidate variance patterns.

For Gamma, divide each variance by the squared mean:

\frac{1{,}000}{100^2}=0.10,\qquad \frac{8{,}000}{200^2}=0.20,\qquad \frac{64{,}000}{400^2}=0.40.

The Gamma ratios double as the mean doubles, rather than remaining stable. For inverse Gaussian, divide by the cubed mean:

\frac{1{,}000}{100^3}=0.001,\qquad \frac{8{,}000}{200^3}=0.001,\qquad \frac{64{,}000}{400^3}=0.001.

The inverse Gaussian dispersion estimate is consistently $\phi=0.001$, corresponding to shape parameter $\lambda=1/\phi=1{,}000$. Its predicted variances are $0.001(100^3)=1{,}000$, $0.001(200^3)=8{,}000$, and $0.001(400^3)=64{,}000$. Each matches the observed variance.

The conditional coefficients of variation also show the distinction. The observed values are $\sqrt{1{,}000}/100\approx0.3162$, $\sqrt{8{,}000}/200\approx0.4472$, and $\sqrt{64{,}000}/400=0.6325$. They increase with mean, as the inverse Gaussian model allows, rather than staying constant as under Gamma.

Check. Each inverse Gaussian ratio equals 0.001, and the implied predictions match the three variances. The common mistake is to infer that Gamma is appropriate merely because all variances are positive and increase with the mean; the diagnostic concerns how quickly they increase.

Use Fitted Residuals as a Focused Variance Check

Grouped summaries are not always available, and a fitted model may be assessed using its residuals. For a severity observation $y_i$ with fitted mean $\widehat{\mu}_i$, a Pearson residual standardized using a candidate variance function is:

r_i=\frac{y_i-\widehat{\mu}_i} {\sqrt{\widehat{\phi}\,V(\widehat{\mu}_i)}}, \qquad V(\mu)= \begin{cases} \mu^2 & \text{Gamma},\\ \mu^3 & \text{inverse Gaussian}. \end{cases}

If the conditional mean and variance are both represented adequately, squared standardized residuals should not show a systematic trend with fitted mean. For a group of observations with similar fitted means, the average squared residual is related to the observed within-group variance divided by the model’s predicted variance. A persistent rise or fall across fitted-mean levels can therefore signal that a candidate variance function is scaling incorrectly.

This is a focused variance check, not a complete assessment of a severity model. First consider whether the mean structure is plausible: an incorrect fitted mean can also produce structured residuals. Also, squared residuals are noisy, and a few unusually large claims can have substantial influence. Residual patterns should be interpreted together with the data definition and the intended conditional target.

Worked Example: Comparing Variance Residual Patterns

A hypothetical model produces three groups of 100 independent uncensored severities with fitted means $80, $160, and $320. The observed within-group variances are $320, $2,560, and $20,480. Suppose the fitted means are considered adequate. Compare the models using an illustrative equal-group average of the three dispersion ratios for each candidate; this simple average is a diagnostic summary, not a claim about a particular GLM fitting procedure.

For Gamma, the ratios are:

\frac{320}{80^2}=0.05,\qquad \frac{2{,}560}{160^2}=0.10,\qquad \frac{20{,}480}{320^2}=0.20.

Their equal-group average is $(0.05+0.10+0.20)/3=0.1167$, rounded to four decimal places. The corresponding predicted variances are approximately $0.1167(80^2)=746.67$, $0.1167(160^2)=2{,}986.67$, and $0.1167(320^2)=11{,}946.67$. The observed-to-predicted variance ratios are approximately $0.4286$, $0.8571$, and $1.7143$. The standardized squared-residual pattern would tend to increase with fitted mean under this Gamma variance choice.

For inverse Gaussian, the ratios are:

\frac{320}{80^3}=0.000625,\qquad \frac{2{,}560}{160^3}=0.000625,\qquad \frac{20{,}480}{320^3}=0.000625.

The inverse Gaussian dispersion is $\phi=0.000625$, so $\lambda=1/\phi=1{,}600$. Its predicted variances are $0.000625(80^3)=320$, $0.000625(160^3)=2{,}560$, and $0.000625(320^3)=20{,}480$. Thus the observed-to-predicted variance ratios are all 1.0000, and this variance pattern is better supported by the summaries.

Check. Under the illustrative calculations, the inverse Gaussian variance matches every group, while Gamma underpredicts variation at the highest fitted mean relative to the lowest. The common mistake is to regard a single dispersion estimate as proof of fit; here the trend across fitted means is the evidence being examined.

Practical Selection and Exam Traps

A disciplined comparison keeps the mean model, response definition, and data basis fixed while examining the variance pattern. If the candidate distributions are fit to different targets, such as one fit to paid amounts and another to an uncensored loss measure, their variance diagnostics are not an apples-to-apples comparison. Similarly, an apparent cubic pattern can be created by pooling groups with unmodeled differences or by a changing deductible or limit mechanism.

When the summaries support one pattern but the overall likelihood or other fit evidence points elsewhere, investigate rather than mechanically selecting by a ratio. The Gamma and inverse Gaussian distributions differ beyond their variance functions, so a variance diagnostic is evidence about model choice, not a complete distributional test. A log-link fit can give similar mean predictions under either family while the implied variances differ substantially.

Do not compare raw variances without scaling. The relevant comparisons are $s^2/\widehat{\mu}^2$ and $s^2/\widehat{\mu}^3$.
Do not treat the two dispersion parameters as interchangeable. Gamma $\phi$ is dimensionless when severity is measured in dollars; inverse Gaussian $\phi$ has units reciprocal to severity. Changing units changes the numerical inverse Gaussian dispersion.
Do not use pooled variance as conditional variance. Mixing groups with different means adds between-group variation and may distort either pattern.
Do not conclude from small groups alone. Sampling variability can make either ratio appear unstable, so the group sizes and extreme observations matter.

Key takeaway: Gamma implies conditional variance proportional to the square of mean severity and constant conditional coefficient of variation. Inverse Gaussian implies variance proportional to the cube and an increasing coefficient of variation. Compare the appropriate scaled variances across comparable groups, then use fitted residual patterns as supporting evidence rather than as a stand-alone decision rule.

Check Your Understanding

For each problem, compare the variance pattern using the stated conditional summaries. Answers are rounded only where needed.

A hypothetical book has two comparable severity groups with means $150 and $300 and variances $900 and $3,600. Calculate both models’ dispersion ratios in each group and identify the better-supported pattern. (Answer: Gamma; the Gamma ratios are 0.0400 and 0.0400, while the inverse Gaussian ratios are 0.0002667 and 0.0001333.)
Two groups have means $50 and $100. Their variances are $250 and $2,000. Calculate the Gamma and inverse Gaussian ratios and state the better-supported pattern. (Answer: Inverse Gaussian; the Gamma ratios are 0.1000 and 0.2000, while the inverse Gaussian ratios are 0.002000 and 0.002000.)
An inverse Gaussian fit has dispersion $\phi=0.0008$ when severity is measured in dollars. What is its shape parameter $\lambda$, and what dispersion results if the same severities are instead measured in thousands of dollars? (Answer: $\lambda=1{,}250$ in the dollar parameterization; the dispersion in thousands of dollars is $0.8$.)
A fitted Gamma model has $\widehat{\phi}=0.09$. For two groups with fitted means $100 and $200, calculate the predicted variances and coefficients of variation. (Answer: Predicted variances are 900 and 3,600; both coefficients of variation are 0.3000.)
A candidate inverse Gaussian model has fitted means $80 and $160 and dispersion $\phi=0.0005$. The observed group variances are $256 and $2,048. Calculate predicted variances and observed-to-predicted ratios, then assess the variance pattern. (Answer: Predicted variances are 256 and 2,048; both ratios are 1.0000, supporting the inverse Gaussian variance pattern.)

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