Inverse Gaussian GLMs for Claim Severity

Why Specify an Inverse Gaussian Severity Model?

A claim severity model describes the distribution of the amount associated with an individual claim, conditional on the claim’s observed characteristics. Choosing a distribution is not just a matter of fitting a curve: its assumed relationship between the mean and variance determines how the model represents variation across risks. The inverse Gaussian distribution is one candidate when severities are positive and variability grows rapidly as the expected severity increases.

The inverse Gaussian GLM is not appropriate merely because claim amounts are positive. Its variance pattern should be plausible for the data being modeled, and observations should correspond to the severity target the model is intended to describe. In particular, a model for individual claim severities is not automatically a model for policy-level aggregate losses.

Definition: An inverse Gaussian GLM for positive claim severities assumes that, conditional on covariates $x_j$, each observed claim severity $Y_j$ is positive and follows an inverse Gaussian distribution with conditional mean $\mu_j>0$ and dispersion $\phi>0$. A link function $g$ relates the mean to the linear predictor: $g(\mu_j)=x_j^\mathsf{T}\beta$. In the usual specification, observations are conditionally independent and share a common dispersion parameter.

For this parameterization, the conditional density at $y>0$ is

f(y\mid \mu,\phi)=\frac{1}{\sqrt{2\pi\phi y^3}} \exp\left(-\frac{(y-\mu)^2}{2\phi\mu^2y}\right).

Here $\mu$ is the mean and $\phi$ is the dispersion parameter. The inverse Gaussian distribution has support only on positive values. Its variance is not simply $\phi$: it depends on both $\phi$ and the mean.

Mean, Variance, and Dispersion

The central specification result is that the variance increases as the cube of the conditional mean:

E[Y\mid x]=\mu,\qquad \operatorname{Var}(Y\mid x)=\phi\mu^3.

Equivalently, the GLM variance function is $V(\mu)=\mu^3$, with dispersion multiplying that function. Thus, for two groups with the same $\phi$, a group whose mean severity is twice as large has a variance eight times as large. This is a much faster increase in variance than the quadratic mean relationship in a Gamma GLM.

The coefficient of variation gives another useful view of the pattern:

\operatorname{CV}(Y\mid x) =\frac{\sqrt{\operatorname{Var}(Y\mid x)}}{\mu} =\sqrt{\phi\mu}.

With common dispersion, the coefficient of variation increases with the mean. This can be a reasonable feature when higher-mean risks also show greater relative variation. It can be implausible when relative variation is roughly constant across the range of fitted means, or when observed variability grows much more slowly or rapidly than the cubic pattern suggests.

Links and the Mean Predictor

A GLM requires a link that maps a positive mean to a predictor. The log link is often convenient for severity modeling because any real-valued linear predictor produces a positive fitted mean:

\log(\mu_j)=x_j^\mathsf{T}\beta, \qquad \mu_j=\exp(x_j^\mathsf{T}\beta).

The log link therefore respects the positive mean domain without imposing a separate restriction on the linear predictor. Coefficients can be interpreted through changes in the fitted mean, but coefficient interpretation and relativity construction are separate tasks from specifying the inverse Gaussian variance pattern.

The canonical link follows from the inverse Gaussian exponential-family representation. With the density parameterization above, the natural parameter is $\theta=-1/(2\mu^2)$. Consequently, the canonical link is

g(\mu)=-\frac{1}{2\mu^2}.

Because this link takes only negative values, a canonical-link fit must keep every linear predictor negative. Its inverse is $\mu=(-1/(2\eta))^{1/2}$, defined for $\eta<0$. This restriction can make the canonical link less convenient than the log link in applications. Sign or scale conventions for the canonical link may be written differently, but the chosen inverse must produce positive means and be used consistently.

Worked Example: Calculating Severity Variance and Relative Variation

A hypothetical book of individual equipment claims is modeled with an inverse Gaussian GLM. For a particular risk profile, the fitted mean severity is $\mu=\$400$, and the fitted dispersion is $\phi=0.0005$ per dollar. Calculate the conditional variance, standard deviation, and coefficient of variation. No interest or mortality assumptions enter this severity calculation.

Method and assumptions. Use the inverse Gaussian variance relationship $\operatorname{Var}(Y\mid x)=\phi\mu^3$, with claim severity measured in dollars and common fitted dispersion $\phi=0.0005$ per dollar.

First cube the mean:

\mu^3=400^3=400(400)(400)=64{,}000{,}000.

Then multiply by dispersion:

\operatorname{Var}(Y\mid x)=0.0005(64{,}000{,}000) =32{,}000\text{ dollars}^2.

The standard deviation is $\sqrt{32{,}000}=\$178.8854\ldots$, and the coefficient of variation is $178.8854\ldots/400=0.447213\ldots$. Rounded only at the final step, to the nearest cent for the standard deviation and four decimals for the coefficient of variation, the results are $\$178.89$ and $0.4472$. The check $\sqrt{\phi\mu}=\sqrt{0.0005(400)}=\sqrt{0.2}=0.4472$ agrees with the directly calculated coefficient of variation.

Common mistake avoided: Do not use the Gamma variance formula $\phi\mu^2$. That incorrect calculation would give $0.0005(400^2)=\$80$, with inconsistent variance units under the stated dispersion. The inverse Gaussian calculation uses $\mu^3$, giving $\$32{,}000$ dollars squared.

Why the Variance Is Cubic

The mean-variance relationship can be justified directly from the inverse Gaussian moment-generating function. Under the density parameterization given above, let $K(t)$ denote the logarithm of the moment-generating function. For $t<1/(2\phi\mu^2)$,

M_Y(t)=\exp(K(t)),\qquad K(t)=\frac{1}{\phi\mu} \left(1-\sqrt{1-2\phi\mu^2t}\right).

The first derivative of $K(t)$ is

K'(t)=\frac{\mu}{\sqrt{1-2\phi\mu^2t}}.

At zero, $K'(0)=\mu$, the mean. Differentiating again gives

K''(t)=\phi\mu^3(1-2\phi\mu^2t)^{-3/2}.

The second derivative of the log moment-generating function at zero is the variance, so $K''(0)=\phi\mu^3$. This establishes both the mean and variance relationship used in the GLM specification.

Worked Example: Applying a Log Link to Claim Characteristics

A hypothetical insurer models positive repair-claim severities using a log-link inverse Gaussian GLM. The specified predictor is $\log(\mu)=\log(200)+(\log 1.5)x_1+(\log 1.1)x_2$, where $x_1=1$ indicates a specialized part and $x_2=2$ is a coded repair-complexity measure. The dispersion is $\phi=0.001$ per dollar. Find the fitted mean, variance, and coefficient of variation. No interest or mortality assumptions enter.

Method and assumptions. The conditional severity is inverse Gaussian with the given mean and dispersion; calculate the mean by exponentiating the predictor, then use the cubic variance function.

Substitute the covariates into the predictor:

\log(\mu)=\log(200)+\log(1.5)+2\log(1.1).

Exponentiating converts the sum of logarithms to a product:

\mu=200(1.5)(1.1)^2 =200(1.5)(1.21) =\$363.

The conditional variance is

\operatorname{Var}(Y\mid x)=0.001(363^3) =0.001(47{,}832{,}147) =47{,}832.147\text{ dollars}^2.

The coefficient of variation is $\sqrt{0.001(363)}=\sqrt{0.363}=0.6024948\ldots$. Rounded at the final step, the fitted mean is $\$363.00$, the variance is $\$47{,}832.15$ dollars squared, and the coefficient of variation is $0.6025$. As a check, the variance is positive and the coefficient of variation is less than one, consistent with the calculated standard deviation being less than the mean.

Common mistake avoided: The predictor is on the log-mean scale, so the fitted mean is the exponential of the entire predictor. Adding the coefficients to 200, rather than exponentiating and multiplying the factors, would not implement the specified model.

Assessing Whether the Variance Pattern Is Plausible

A cubic variance function is a modeling assumption, not a guarantee that observed data will follow it exactly. A useful diagnostic idea is to compare severity variation across groups or fitted-mean ranges after accounting for the model’s covariates. If the inverse Gaussian pattern is appropriate with a common dispersion, the ratios of conditional variance to cubed conditional mean should be broadly similar:

\frac{\operatorname{Var}(Y\mid x)}{\mu^3}\approx\phi.

In actual samples, group variances fluctuate, especially when groups contain few claims. Therefore, the ratios need not match exactly. Large systematic differences may indicate that a single common dispersion and cubic variance function do not adequately represent the data. A fitted mean relationship alone is not enough to validate the distributional specification.

The model is most plausible when the target consists of positive individual claim severities, the inverse Gaussian distribution provides a reasonable representation of their conditional distributions, and the observed spread increases roughly in line with $\mu^3$. It may be unsuitable if the response includes zero amounts, if the sampling process produces a selected or censored severity distribution, or if the conditional variance pattern strongly conflicts with the cubic relationship. Such data features require an explicit treatment; they should not be assumed to disappear simply by choosing an inverse Gaussian GLM.

Worked Example: Checking a Common Dispersion Pattern

For a hypothetical portfolio of positive machinery-repair claims, three covariate groups have fitted mean severities of $\$100$, $\$200$, and $\$400$. Their illustrative within-group severity variances are $5{,}000$, $40{,}000$, and $320{,}000$ dollars squared, respectively. Assess whether these summaries are consistent with a common-dispersion inverse Gaussian model, and calculate the implied coefficient of variation in each group. Treat the supplied variances as the group variances for this calculation; no interest or mortality assumptions enter.

Method and assumptions. For each group, divide its variance by its mean cubed. A common-dispersion inverse Gaussian model implies the same ratio $\phi$ in every group.

For the first group, $\mu^3=100^3=1{,}000{,}000$, so $\phi=5{,}000/1{,}000{,}000=0.005$. For the second group, $\mu^3=200^3=8{,}000{,}000$, so $\phi=40{,}000/8{,}000{,}000=0.005$. For the third group, $\mu^3=400^3=64{,}000{,}000$, so $\phi=320{,}000/64{,}000{,}000=0.005$.

All three ratios agree, so these summaries are exactly consistent with a common dispersion of $0.005$. The group coefficients of variation are $\sqrt{0.005(100)}=\sqrt{0.5}=0.7071$, $\sqrt{0.005(200)}=1.0000$, and $\sqrt{0.005(400)}=\sqrt{2}=1.4142$. The relative variation rises with the mean, as the inverse Gaussian model predicts. This check supports the variance pattern in these summaries, but does not by itself establish that the full distribution is appropriate.

Common mistake avoided: Comparing the raw variances alone would miss the scale implied by the different means. The relevant comparison is variance divided by mean cubed, not variance divided by mean or by mean squared.

Practical Checks and Exam Traps

When specifying this model, state the response, its sampling unit, the conditional mean structure, the link, and the dispersion-based variance relationship. These details make clear what the fitted distribution claims about the data. For a claim-level severity model, the response should be the positive severity for each sampled claim, not a claim count or a policy’s total loss.

Do not interchange variance functions. Inverse Gaussian variance is $\phi\mu^3$; Gamma variance is proportional to $\mu^2$.
Keep dispersion units consistent. If severity is measured in dollars, $\phi$ has units of inverse dollars so that $\phi\mu^3$ has units of dollars squared.
Check the link’s domain. A log link yields positive means for all real predictors; the canonical link requires a negative predictor.
Do not infer distributional fit from a good mean fit alone. The conditional variance pattern and the positive continuous response assumption also matter.

These specifications are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. In an applied analysis, the inverse Gaussian model is a candidate whose assumptions must be evaluated against the claim-severity data and the target being modeled.

Check Your Understanding

Work each problem using the inverse Gaussian mean-variance relationship and the stated assumptions.

A hypothetical log-link severity model has $\log(\mu)=\log(180)+x\log(1.25)$, with $x=2$, and $\phi=0.002$ per dollar. Calculate the fitted mean and variance. (Answer: $\mu=\$281.25$; variance $=\$44{,}494.63$ dollars squared.)
A positive claim severity has fitted mean $\$250$ and inverse Gaussian dispersion $0.004$ per dollar. Calculate its variance and coefficient of variation. (Answer: variance $=\$62{,}500$ dollars squared; coefficient of variation $=1.0000$.)
Two hypothetical groups have means $\$80$ and $\$160$, and variances $5{,}120$ and $40{,}960$ dollars squared. Calculate each implied dispersion and assess whether a common-dispersion inverse Gaussian variance pattern is consistent with these summaries. (Answer: both implied dispersions are $0.0100$, so the summaries are consistent with a common-dispersion pattern.)
Under the canonical link $g(\mu)=-1/(2\mu^2)$, calculate the predictor for $\mu=\$200$. Then find the mean corresponding to predictor $\eta=-0.00002$. (Answer: the predictor is $-0.0000125$; the mean is $\sqrt{25{,}000}=\$158.11$, rounded to cents.)
A hypothetical portfolio shows that the conditional coefficient of variation is approximately constant as fitted mean severity increases substantially. Is that pattern, by itself, strong support for a common-dispersion inverse Gaussian model? Explain using its variance relationship. (Answer: No. With common $\phi$, the inverse Gaussian coefficient of variation is $\sqrt{\phi\mu}$, so it increases with the mean rather than remaining constant.)

What You'll Learn