Why the Meaning of a Weight Matters
A severity GLM may assign a positive number to each observation, but that number is not self-explanatory. It might say that a record stands for several identical claims, that an observation is measured more precisely than another, or that a sampled claim represents many claims in the portfolio. These interpretations are not interchangeable. Using the wrong one can change the fitted target or make the reported uncertainty misleading.
The response in this tutorial is a claim-level severity \(Y_i\), such as the amount paid for claim \(i\), conditional on its rating variables \(x_i\). Earlier material on claim-level severity targets distinguishes the unit being modeled from the policy or exposure that generated the claim. Here the question is narrower: what does a weight attached to that observation mean?
The distinction matters even when two weighting schemes happen to produce the same coefficient estimate. That equality can occur in a simple intercept-only Gamma model, but the implied data, target, and uncertainty calculations can still be different.
A Common Severity Model and Its Weighted Score
For the examples, use a Gamma GLM with a log link. Conditional on the rating variables, suppose \(Y_i\) has mean \(\mu_i\), variance \(\phi\mu_i^2\), and density parameterized by shape \(1/\phi\) and scale \(\phi\mu_i\). Thus \(E[Y_i\mid x_i]=\mu_i\), and \(\operatorname{Var}(Y_i\mid x_i)=\phi\mu_i^2\). The log link means \(\log\mu_i=x_i^\mathsf{T}\beta\). Assume independent observations conditional on their covariates unless a design or dependence adjustment is explicitly made. The dispersion \(\phi\) is positive and common to observations in the unweighted model.
A conventional precision-weighted Gamma model specifies \(\operatorname{Var}(Y_i\mid x_i)=\phi\mu_i^2/w_i\), where \(w_i>0\) is a relative precision. A frequency-weighted likelihood instead multiplies the ordinary likelihood contribution for a row by its replication count \(w_i\). Both lead to weighted score contributions in this model, but they express different assumptions.
For sampling weights, the same algebra is often used as a weighted estimating equation or pseudo-likelihood, rather than as a claim that the sampled observations are literal independent replications. The equation alone does not determine which interpretation is appropriate.
For an intercept-only model, all means are equal to \(\mu\). Solving the score equation gives a useful result:
This derivation explains why several kinds of weights can yield the same weighted average in a simple example. It does not mean the weights have the same statistical meaning. In a model with rating variables, weights enter the score across covariate patterns; under model misspecification, they can also change the particular compromise represented by the fitted coefficients.
Three Weighting Schemes
| Weight type | What the row means | What the fit targets | Key caution |
|---|---|---|---|
| Frequency | The row represents the stated number of identical observations at the same response and covariates. | The likelihood fit to the expanded data set of those repeated observations. | Fractional weights are not literal counts of replicated claims. |
| Precision | The observation has known relative precision; in the stated Gamma model its variance is \(\phi\mu_i^2/w_i\). | A conditional mean relationship, with more precise observations contributing more information. | A large weight does not assert that more claims occurred. |
| Sampling | The sampled record represents members of a target population according to the inclusion design. | A population-oriented estimating target, when the weights and model are appropriate. | Weighted model-based standard errors alone may not account for the sampling design. |
A frequency weight is most straightforward when identical records have been compressed for storage: expanding a row with integer weight \(k\) produces \(k\) observations with the same response and covariates. A precision weight is appropriate only when the variance or information interpretation is justified. A sampling weight is commonly \(1/\pi_i\), where \(\pi_i\) is the inclusion probability; it corrects representation, not measurement noise.
All three can affect fitted coefficients. If the conditional mean model is correctly specified and sampling is ignorable conditional on modeled variables, changing observation weights may leave the same population mean relationship as the target. If the model is misspecified, different weights generally emphasize different portions of the data and can yield different fitted coefficients. State the target and assumptions rather than asserting that weights never change the estimand.
Worked Example: Expanding Frequency-Weighted Claim Records
A hypothetical portfolio has five claim-level observations at the same rating level. To save space, the data are stored as two rows: severity \(1{,}000\) dollars with frequency weight 3, and severity \(1{,}600\) dollars with frequency weight 2. Fit the common mean in an intercept-only Gamma log-link model. Assume independent claims, \(\phi=0.25\) (shape \(1/\phi=4\), scale \(\phi\mu\)); interest and mortality assumptions do not apply.
Method. Treat each integer weight as the number of identical claim records in the expanded likelihood. The common mean estimate is the frequency-weighted mean.
The total represented claim count is \(3+2=5\). The weighted severity total is \(3(1{,}000)+2(1{,}600)=3{,}000+3{,}200=6{,}200\) dollars. Therefore,
Check. Expanding the rows gives \(1{,}000,1{,}000,1{,}000,1{,}600,1{,}600\), whose arithmetic mean is \(6{,}200/5=1{,}240\) dollars. The common mistake this avoids is treating the two stored rows as only two claims and reporting \((1{,}000+1{,}600)/2=1{,}300\) dollars.
Worked Example: Using Relative Precision, Not Claim Counts
Suppose two hypothetical severity measurements at the same rating level are \(800\) and \(1{,}200\) dollars. Their relative precision weights are 1 and 3, respectively: the second measurement is assumed to have three times the precision of the first. Use a Gamma log-link intercept-only model with \(\operatorname{Var}(Y_i\mid x_i)=\phi\mu^2/w_i\), \(\phi=0.25\), and independent measurements under this model. For each observation, the implied Gamma shape is \(w_i/\phi\) and scale is \(\phi\mu/w_i\). Interest and mortality assumptions do not apply.
Method. Apply the weighted score for the stated precision model. The weight changes each observation's contribution to estimating the common mean; it does not turn the second measurement into three claims.
The weighted total is \(1(800)+3(1{,}200)=800+3{,}600=4{,}400\) dollars. The sum of weights is \(1+3=4\). Thus,
Without the precision information, the unweighted mean would be \((800+1{,}200)/2=1{,}000\) dollars. The estimate of \(1{,}100\) uses the specified relative precision. The common mistake is to describe this as four observed claims, or to claim that a weight of 3 necessarily means three independent claim severities were collected. Neither follows from the precision assumption.
Worked Example: Correcting for Unequal Sampling Probabilities
A hypothetical insurer draws a sample of two claims from a target claim population. A claim with severity \(900\) dollars had inclusion probability \(0.50\); a claim with severity \(1{,}500\) dollars had inclusion probability \(0.25\). Use inverse-probability sampling weights in an intercept-only Gamma log-link estimating equation to estimate the population mean severity. Assume the conditional severity model has \(\phi=0.25\), shape \(1/\phi=4\), and scale \(\phi\mu\); the weights here arise from sampling, not from that Gamma variance. Interest and mortality assumptions do not apply.
Method. Set each sampling weight to the reciprocal of its inclusion probability. The two weights are \(1/0.50=2\) and \(1/0.25=4\). Use the weighted estimating mean, which is a ratio of the weighted severity total to the total weight.
The weighted severity total is \(2(900)+4(1{,}500)=1{,}800+6{,}000=7{,}800\) dollars, and the total weight is \(2+4=6\). Therefore,
The unweighted sample mean is \((900+1{,}500)/2=1{,}200\) dollars, but it gives equal representation to observations with unequal inclusion probabilities. The weighted result is the inverse-probability weighted estimate for this sample. It is not a guarantee that this particular small sample exactly recovers the population mean. Its common trap is to treat weights 2 and 4 as precision values and claim a variance of \(\phi\mu^2/2\) and \(\phi\mu^2/4\); that variance interpretation was not given.
Uncertainty, Diagnostics, and Practical Decisions
Weighting affects more than the point estimate. Under a frequency-weight interpretation, the data contain the represented number of observations, so likelihood information and degrees of freedom reflect the expanded data, subject to the usual independence and model assumptions. Under a precision-weight model, the stated variance relationship determines how information varies across observations. For sampling weights, the design can induce dependence or unequal selection probabilities; a naive model-based covariance matrix may therefore understate or otherwise misstate uncertainty. Design-based variance estimation or a justified robust method may be needed.
The dispersion estimate also depends on the chosen interpretation and estimation procedure. Do not assume that multiplying score contributions by weights automatically gives a valid dispersion estimate or standard error for every purpose. In particular, an inverse inclusion probability is not a precision estimate, and a count of replicated records is not an inverse inclusion probability.
Before fitting a severity GLM, answer these questions:
Identify whether each row is one claim, a repeated record, or a sampled claim representing a larger population.
Give a replication count, a variance/precision assumption, or an inclusion probability and its inverse. Do not rely on a software argument named merely “weight.”
Specify whether the objective is the expanded-sample likelihood, a conditional mean fit with unequal precision, or a population-oriented fit under a sampling design.
Check that the variance calculation follows the assumed likelihood, precision structure, or sampling design rather than borrowing one interpretation's standard errors for another.
A common exam trap is to infer the weight's meaning from the software syntax alone. Implementations differ in how they label frequency, prior, analytic, and sampling weights. Translate the intended model into its likelihood or estimating equation, then check that the software's weight option implements that equation. The meaning of the weight, not its column heading, determines the fitted target and the appropriate uncertainty calculation.
Check Your Understanding
For each problem, identify the weight interpretation before calculating or describing the fitted result.
- A compressed claim file has severity \(700\) dollars with frequency weight 2 and severity \(1{,}300\) dollars with frequency weight 3. In an intercept-only Gamma model, calculate the expanded-data mean. (Answer: \(1{,}060\) dollars.)
- Two independent severity measurements at one rating level are \(900\) and \(1{,}500\) dollars, with stated precision weights 2 and 1. Under the corresponding precision-weighted intercept-only mean equation, calculate the estimate. (Answer: \(1{,}100\) dollars.)
- A sampled claim of \(800\) dollars has inclusion probability \(0.40\), and a sampled claim of \(1{,}400\) dollars has inclusion probability \(0.20\). Calculate the inverse-probability weighted intercept-only estimate. (Answer: \((2.5(800)+5(1{,}400))/(2.5+5)=1{,}200\) dollars.)
- A row has a severity of \(1{,}100\) dollars and weight 4 because it represents four identical claims. Is it valid to conclude that its variance is one-quarter the variance of a single claim solely from that weight? (Answer: No; the weight represents four replicated records, not a precision assumption.)
- A sample uses inverse inclusion-probability weights in a severity GLM. Does using those weights alone guarantee that ordinary model-based standard errors account for the sampling design? (Answer: No; the variance calculation must also account for the design or use a justified design-robust method.)