Deriving Weights for Grouped Claim Severities

Why Grouped Severity Data Need Weights

Severity data are sometimes stored by cell rather than by claim. A cell might be a combination of rating characteristics, such as vehicle type and territory, with a recorded claim count and average severity. The average is useful, but it is not as precisely measured in a cell containing two claims as in one containing two hundred claims. A grouped severity fit must account for that difference.

The key result is that, under specified conditions, a cell mean based on $n_j$ claims has variance equal to the individual-claim variance divided by $n_j$. Thus its variance is reduced in proportion to the number of claims. In a grouped GLM, this leads to a claim-count prior weight of $n_j$, not $n_j^2$, and not an unweighted fit to cell means.

Definition: For cell $j$, let $n_j$ be the number of observed claims and let $Y_{jk}$ be the severity of claim $k$, for $k=1,\ldots,n_j$. The grouped response is the cell mean $\bar Y_j = \frac{1}{n_j}\sum_{k=1}^{n_j}Y_{jk}$. A claim-count weight is the factor $n_j$ used to account for how many claim-level observations the cell mean represents.

For the derivation and examples below, assume claims within a cell are conditionally independent, have the same modeled mean $\mu_j$, and share the same dispersion parameter $\phi$. Assume the individual severity variance follows the GLM variance function $\operatorname{Var}(Y_{jk}\mid j)=\phi V(\mu_j)$. The cell members must also have the same covariate values, so that the model assigns them the same $\mu_j$. These assumptions matter: a count weight cannot repair dependence or turn claims with different modeled means into identical observations.

Deriving the Claim-Count Weight

The variance of an average of independent observations is the sum of their variances divided by the square of the number averaged. Under the assumptions above:

\operatorname{Var}(\bar Y_j\mid j) =\operatorname{Var}\left(\frac{1}{n_j}\sum_{k=1}^{n_j}Y_{jk}\right) =\frac{1}{n_j^2}\sum_{k=1}^{n_j}\phi V(\mu_j) =\frac{\phi V(\mu_j)}{n_j}.

The same relation can be expressed in the standard GLM variance form: the variance of the grouped response is the usual variance function multiplied by dispersion $\phi/n_j$. Equivalently, it has the same variance function and dispersion $\phi$, with prior weight $n_j$. This is the sense in which the cell count is the grouped-response weight.

Theorem: If all claims in a cell are independent and share the same modeled mean and variance function, then $\operatorname{Var}(\bar Y_j\mid j)=\phi V(\mu_j)/n_j$. A grouped GLM using $\bar Y_j$ with prior weight $n_j$ has the same quasi-score for the model parameters as the claim-level fit. The equality applies to the estimating equations under the stated conditions; it does not mean every grouped likelihood or diagnostic is identical to its claim-level counterpart.

To see why the score agrees, write the GLM linear predictor as $\eta_j$, with link $g(\mu_j)=\eta_j$. In a quasi-likelihood GLM, the contribution to the score for a parameter vector $\beta$ from one claim is proportional to $\frac{Y_{jk}-\mu_j}{\phi V(\mu_j)}\frac{d\mu_j}{d\eta_j}x_j$, where $x_j$ is the covariate vector. Summing over claims in cell $j$ replaces the sum of their residuals by $n_j(\bar Y_j-\mu_j)$. The grouped score with prior weight $n_j$ therefore gives the same contribution:

\sum_{k=1}^{n_j} \frac{Y_{jk}-\mu_j}{\phi V(\mu_j)} \frac{d\mu_j}{d\eta_j}x_j = n_j\frac{\bar Y_j-\mu_j}{\phi V(\mu_j)} \frac{d\mu_j}{d\eta_j}x_j.

Summing this equality across cells shows that the claim-level and grouped quasi-score equations are identical. Their solutions for $\beta$ are therefore the same when the model and data meet the assumptions and the same estimating method is used. This argument also explains why using the number of claims is not an arbitrary convention: the grouped residual is an average, while its weight restores the sum of claim-level residual contributions.

Prior Weights and Working Weights

A claim-count prior weight is not itself the complete weight used at every stage of a GLM calculation. In iteratively reweighted least squares, the working weight also depends on the variance function and link. For grouped response $\bar Y_j$, the working weight is

W_j^{\text{working}} = \frac{n_j}{\phi V(\mu_j)} \left(\frac{d\mu_j}{d\eta_j}\right)^2.

This expression follows from the grouped variance $\phi V(\mu_j)/n_j$ and the local change in the mean with the linear predictor. The count $n_j$ is the prior weight; $V(\mu_j)$, $\phi$, and the link determine how that prior weight translates into the working weight. For a Gamma GLM, $V(\mu)=\mu^2$; for an inverse Gaussian GLM, $V(\mu)=\mu^3$. With a log link, $d\mu/d\eta=\mu$, so the working weight is $n_j/\phi$ for Gamma and $n_j/(\phi\mu_j)$ for inverse Gaussian.

Some software accepts prior or case weights; other procedures request a dispersion adjustment or a working weight. Those options are not interchangeable without checking the software’s definition. In particular, entering an inverse-variance weight that already includes $n_j$ and also applying a claim-count weight can count the claims twice.

Worked Example: Deriving a Gamma Cell’s Precision

A hypothetical severity cell contains $n=25$ independent claims. Suppose each claim is modeled with a Gamma distribution having mean $\mu=1{,}000$ dollars and dispersion $\phi=0.16$, so that its variance is $\phi\mu^2$. Find the variance and standard deviation of the cell mean, and state its grouped prior weight. Interest and mortality assumptions are not applicable to this severity calculation.

Method and assumptions. Apply the grouped-mean variance formula. All 25 claims are assumed to have the same modeled mean and dispersion, and to be conditionally independent.

First calculate the individual-claim variance:

\operatorname{Var}(Y)=0.16(1{,}000)^2 =0.16(1{,}000{,}000) =160{,}000\text{ dollars}^2.

Divide by the claim count to obtain the variance of the average:

\operatorname{Var}(\bar Y)=\frac{160{,}000}{25} =6{,}400\text{ dollars}^2.

The standard deviation is $\sqrt{6{,}400}=80$ dollars, and the prior weight is $n=25$. As a check, the grouped Gamma variance formula gives $0.16(1{,}000)^2/25=6{,}400$, as required. The common mistake is to use the individual variance as the variance of the mean; that would overstate the grouped variance by a factor of 25.

When the Grouped Fit Matches the Claim-Level Fit

A grouped fit matches the claim-level fit for model coefficients when the claims in each cell share the same design row and modeled mean, the independence and common-variance assumptions hold, and the grouped procedure uses the claim count as its prior weight. The score derivation establishes this for quasi-likelihood estimation. For standard Gamma and inverse Gaussian GLMs with a common dispersion parameter, the mean-parameter score also reduces to a function of the cell count and cell sum; the grouped mean retains that sum. Thus grouping can preserve the coefficient fit under these conditions.

This conclusion has limits. If claims within a cell have different covariates, then they generally have different modeled means, and the cell average alone does not reproduce the individual score. If claims are dependent, the variance of the average need not decrease by exactly $1/n_j$. Also, retaining only cell means discards within-cell variation. That can affect dispersion estimation, residual analysis, and other diagnostics even when the coefficient estimates agree. For a distribution or fitting method that does not preserve the relevant likelihood information under aggregation, equality of quasi-scores should not be mistaken for equality of full likelihoods.

Check the cell definition.
Every claim in a cell should have the same modeled covariates and target severity definition.

Check the sampling assumptions.
Claims should be conditionally independent and share the cell’s modeled mean and variance structure.

Use the count as the prior weight.
For a grouped mean based on $n_j$ claims, use $n_j$, then let the GLM variance function and link determine the working weight.

Check what grouping has removed.
Do not assume cell means preserve within-cell dispersion information or every feature of the full likelihood.

Worked Example: Matching a Gamma Intercept Fit

A hypothetical portfolio has two severity cells. Cell A has $n_A=2$ claims with mean severity $\bar Y_A=100$ dollars; cell B has $n_B=3$ claims with mean severity $\bar Y_B=200$ dollars. Fit a Gamma GLM with a log link and intercept only, using common dispersion $\phi=0.2$. Find the fitted mean and verify that grouping gives the same score equation as the five claim-level observations. Assume claims are independent and cell membership does not change the intercept-only mean. Interest and mortality assumptions are not applicable.

Method and assumptions. With an intercept-only model, every claim has the same fitted mean $\mu$. Under a Gamma variance function $V(\mu)=\mu^2$ and log link, $d\mu/d\eta=\mu$. The claim-level score is proportional to $\sum_k(Y_k-\mu)/(\phi\mu)$.

The total number of claims is $2+3=5$, and their total severity is

2(100)+3(200)=200+600=800\text{ dollars}.

The claim-level score equation is therefore

\frac{800-5\mu}{0.2\mu}=0, \qquad 800-5\mu=0, \qquad \hat\mu=160\text{ dollars}.

The grouped score, with prior weights 2 and 3, gives the same result:

\frac{2(100-\mu)+3(200-\mu)}{0.2\mu}=0, \qquad 800-5\mu=0, \qquad \hat\mu=160\text{ dollars}.

At $\hat\mu=160$, the grouped residual numerator is $2(-60)+3(40)=-120+120=0$, verifying the solution. If the two cell means were averaged without weights, the result would be $(100+200)/2=150$ dollars. That is the common mistake: it gives each cell equal influence despite the different claim counts. The correctly weighted fit equals the claim-level arithmetic mean, $800/5=160$ dollars.

Worked Example: Unequal Cell Counts and Gamma Working Weights

A hypothetical portfolio has two independent Gamma severity cells with a common dispersion $\phi=0.25$. Cell A contains 4 claims with mean severity $550$ dollars; cell B contains 9 claims with mean severity $480$ dollars. An intercept-only log-link model is fitted. Calculate the fitted mean, then calculate the working weight for each cell at that fitted mean. Interest and mortality assumptions are not applicable.

Method and assumptions. With a Gamma variance function and log link, the grouped score gives the claim-count-weighted arithmetic mean, and the working weight is $n_j/\phi$. Assume the stated cell means summarize claims with common modeled means within each cell.

The total severity represented is $4(550)+9(480)=2{,}200+4{,}320=6{,}520$ dollars, and the claim count is $4+9=13$. Hence

\hat\mu=\frac{6{,}520}{13} =501.538\ldots\text{ dollars}.

Rounded to the nearest cent, $\hat\mu=\$501.54$. For Gamma with a log link, $V(\mu)=\mu^2$ and $d\mu/d\eta=\mu$, so

W_j^{\text{working}} =\frac{n_j}{0.25\mu^2}\mu^2 =\frac{n_j}{0.25}.

Thus cell A’s working weight is $4/0.25=16$, and cell B’s is $9/0.25=36$. Their ratio, $16:36=4:9$, matches the ratio of their claim counts. The weighted fitted mean checks out because $4(550-501.538\ldots)+9(480-501.538\ldots)=193.846\ldots-193.846\ldots=0$. The common error is to use weights proportional to $n_j^2$; the variance of the mean contains $1/n_j$, so its inverse contributes $n_j$, not $n_j^2$.

Common Mistakes and Practical Checks

Giving every cell equal weight. An unweighted fit treats a cell based on one claim like a cell based on one hundred claims. Use the claim count when the assumptions for grouped means hold.
Squaring the count. The variance of a mean divides by $n_j$; its precision therefore grows in proportion to $n_j$, not $n_j^2$.
Confusing prior and working weights. The claim-count prior weight is $n_j$. The full working weight also depends on $\phi$, the variance function, and the link.
Grouping claims with different modeled means. A single cell mean generally cannot preserve the individual score when its claims have different covariates.
Assuming all diagnostics are preserved. Cell means do not reveal the spread of severities within each cell, so dispersion assessment and residual checks may require claim-level information.

These weighting principles are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. In an exam calculation, state the variance assumption, derive the grouped variance, identify the prior weight, and check whether the cell construction permits a claim-level equivalence.

Check Your Understanding

For each problem, state the relevant assumption and show how the claim count enters the grouped calculation.

A hypothetical Gamma severity cell has $n=16$, mean $\mu=800$ dollars, and dispersion $\phi=0.09$, with independent claims sharing the same mean. Find the variance and standard deviation of the cell mean, and its prior weight. (Answer: variance $3{,}600$ dollars squared; standard deviation $\$60$; prior weight $16$.)
Two independent Gamma cells have $n_1=3,\bar Y_1=\$400$ and $n_2=7,\bar Y_2=\$520$. Under a common-mean intercept-only log-link model, find the grouped fitted mean. (Answer: $\$484$.)
For a Gamma model with $\phi=0.20$, log link, and fitted mean $\mu=\$390$, find the grouped variance of the mean for cells with $n=5$ and $n=15$. (Answer: $6{,}084$ and $2{,}028$ dollars squared, respectively.)
Two independent Gamma cells have counts 2 and 5, with observed means $\$90$ and $\$150$. Under a common-mean intercept-only log-link model, calculate the fitted mean and explain why the unweighted average is not the grouped fit. (Answer: fitted mean $\$132.86$; the unweighted average $\$120$ incorrectly gives the two cells equal influence.)
A cell combines claims from two distinct covariate groups, each with a different modeled mean. Is using the total claim count as the weight on the combined cell mean guaranteed to reproduce the claim-level score? (Answer: No; the claims do not share a modeled mean, so the combined mean and total count generally do not preserve the claim-level score.)

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