Severity Models for Audited or Sampled Claims

Why Audited Claims May Not Represent the Portfolio

A claim file assembled for audit, review, or research may contain claims selected at different rates. For example, a hypothetical insurer might review large claims more often than small ones, or sample claims differently across risk classes. The resulting file can be useful, but its unadjusted severity distribution describes the selected claims, not necessarily the insurer’s target claim population.

This distinction matters when estimating a portfolio average, describing the severity distribution, or fitting a model intended to represent the portfolio. If a claim’s chance of appearing in the data depends on its severity, an unweighted analysis can overrepresent or underrepresent parts of the severity range. Known inclusion probabilities provide a way to correct that imbalance, subject to important conditions.

Definition: Let the target population consist of claims $i=1,\ldots,N$, with severity $Y_i$ and, when relevant, claim characteristics $X_i$. Let $R_i=1$ if claim $i$ is included in the audited or sampled data and $R_i=0$ otherwise. The claim’s inclusion probability is $\pi_i=P(R_i=1)$, or more generally $P(R_i=1\mid Y_i,X_i)$ when selection depends on claim information. If $\pi_i$ is known and positive, its inverse $w_i=1/\pi_i$ is its inverse-probability sampling weight.

The target population and the severity response must be specified before weights are applied. For example, if the target is all claims in a period, the response might be claim-level incurred severity at a common valuation date. Weighting changes how much each observed claim represents; it does not change the definition of the response, convert a censored amount into an uncensored amount, or repair an incorrectly specified target.

The central requirement is positivity: every type of claim that occurs in the target population must have a nonzero chance of inclusion. The selection probabilities must also be known or credibly estimated, and the observed severity and covariates must be recorded consistently with the target definition. If high-severity claims are deliberately sampled more often, their weights are smaller, not larger.

How Inverse-Probability Weighting Works

For a finite target population with known claim-level inclusion probabilities, the Horvitz–Thompson estimator of a population total replaces each unobserved population contribution by its observed contribution divided by its probability of observation. For a claim-level quantity $g(Y_i,X_i)$, it is

\widehat{T}_{HT}=\sum_{i:R_i=1}\frac{g(Y_i,X_i)}{\pi_i}.

If the population size $N$ is known, dividing this estimated total by $N$ gives an unbiased design-based estimate of the population mean. If $N$ is not known, a commonly used normalized, or Hájek, estimate divides the weighted total by the sum of the weights:

\widehat{\mu}_{H} = \frac{\displaystyle\sum_{i:R_i=1}Y_i/\pi_i} {\displaystyle\sum_{i:R_i=1}1/\pi_i}.

The normalized estimate is a ratio estimator, so it is not generally exactly unbiased in a finite sample. It is often useful because it estimates the population mean without requiring the population size to be known, and is design-consistent under standard sampling conditions.

Derivation: For a fixed population claim $i$, $R_i$ is an indicator with $E[R_i]=\pi_i$. Therefore $E[R_i g(Y_i,X_i)/\pi_i]=g(Y_i,X_i)$, provided $\pi_i>0$. Summing this equality over the population gives $E[\widehat{T}_{HT}]=\sum_{i=1}^{N}g(Y_i,X_i)$. Thus each observed claim, on average over the selection design, stands in for $1/\pi_i$ population claims.

The same idea estimates the target severity distribution. At a severity value $y$, use $g(Y_i)=\mathbf{1}(Y_i\le y)$, where the indicator is one when the condition holds and zero otherwise. The normalized weighted empirical distribution is

\widehat{F}_{H}(y) = \frac{\displaystyle\sum_{i:R_i=1}\frac{1}{\pi_i}\mathbf{1}(Y_i\le y)} {\displaystyle\sum_{i:R_i=1}\frac{1}{\pi_i}}.

A weighted mean, weighted distribution, and weighted model are different summaries of the same correction principle. A weighted likelihood for a severity model with density or probability function $f(y\mid x;\theta)$ can be written as

\ell_w(\theta) = \sum_{i:R_i=1}\frac{1}{\pi_i}\log f(Y_i\mid X_i;\theta).

This is a weighted, or pseudo-, likelihood contribution for the target population. Its use relies on a suitable severity model and appropriate selection weights; the weighted likelihood is not a substitute for defining the right response or accounting for censoring. Inference must also reflect the sampling design when design-based standard errors are required. First-order inclusion probabilities suffice for the expectation of a total, but variance estimation generally also requires information about joint inclusion probabilities or another appropriate variance method.

Worked Example: Estimating a Mean From Unequally Sampled Claims

A hypothetical audit sample contains four claims with severities of $1,000, $1,000, $3,000, and $5,000. The respective known inclusion probabilities are 0.50, 0.50, 0.25, and 0.50. Treat the claims as exact discrete severity observations at a common valuation date; no parametric severity distribution, interest assumption, or mortality assumption is used. Estimate the target population mean using normalized inverse-probability weights.

Method. Use $w_i=1/\pi_i$, then divide the weighted severity total by the sum of the weights.

Step 1: Calculate the weights. The two $1,000 claims each have weight $1/0.50=2$. The $3,000 claim has weight $1/0.25=4$. The $5,000 claim has weight $1/0.50=2$.

Step 2: Calculate the weighted severity total and total weight.

\sum_i w_iY_i =2(1{,}000)+2(1{,}000)+4(3{,}000)+2(5{,}000) =26{,}000,

\sum_iw_i=2+2+4+2=10.

Step 3: Calculate the normalized estimate. The estimated target mean is $26{,}000/10=\$2{,}600.00$. By comparison, the unweighted sample mean is $(1{,}000+1{,}000+3{,}000+5{,}000)/4=\$2{,}500.00$. Here the lower inclusion probability of the $3,000 claim gives it greater representation in the weighted estimate.

Check. The weighted empirical distribution assigns masses $4/10=0.4$, $4/10=0.4$, and $2/10=0.2$ to $1,000, $3,000, and $5,000. Its mean is $0.4(1{,}000)+0.4(3{,}000)+0.2(5{,}000)=\$2{,}600.00$, agreeing with the calculation above. The common mistake avoided is treating each observed claim as one equally representative claim.

Worked Example: Reconstructing a Discrete Severity Mix

A hypothetical sample has 100 claims: 60 claims have severity $1,000, 30 have severity $3,000, and 10 have severity $9,000. Their known inclusion probabilities are 0.60, 0.30, and 0.10, respectively. Assume each claim within a severity group has the stated group probability of inclusion. Estimate the target severity proportions and mean. The severity distribution is treated as discrete at those three amounts; no parametric distribution or interest or mortality assumptions are needed.

Step 1: Weight each group. The group weights per claim are $1/0.60=5/3$, $1/0.30=10/3$, and $1/0.10=10$. Thus the estimated population counts are $60(5/3)=100$, $30(10/3)=100$, and $10(10)=100$.

Step 2: Find the estimated total population size and severity proportions. The estimated total is $100+100+100=300$ claims. Each severity group therefore has estimated proportion $100/300=1/3$.

Step 3: Calculate the target mean. The estimated weighted severity total is

$$ 100(1{,}000)+100(3{,}000)+100(9{,}000) =1{,}300{,}000. $$

Dividing by 300 gives an estimated mean of $\$4{,}333.33$, rounded to the nearest cent. The unweighted sample mean is $[60(1{,}000)+30(3{,}000)+10(9{,}000)]/100=\$2{,}400.00$. The difference arises because the sample’s high-severity claims were included at a lower rate than the low-severity claims.

Check. The reconstructed probabilities sum to one, and their weighted mean is $[1{,}000+3{,}000+9{,}000]/3=\$4{,}333.33$, to the nearest cent. The common mistake avoided is interpreting the sample proportions, 60%, 30%, and 10%, as the target portfolio proportions.

Selection by Risk Class Versus Selection by Severity

Selection can depend on observed risk characteristics without directly depending on severity within each class. These situations have different implications. If inclusion depends only on $X$, then, under the condition $R$ is independent of $Y$ given $X$, the conditional severity distribution $Y\mid X$ can remain representative within each class. But the sample may contain the wrong mix of classes. An unadjusted overall mean or an intercept-only model can then misrepresent the portfolio. Weighting or standardizing predictions to the target class mix addresses that problem.

If selection depends on severity even after conditioning on recorded risk characteristics, the observed conditional severity distribution can also be distorted. In that case, fitting a conditional severity model without addressing the selection mechanism can bias the estimated severity parameters. Inverse-probability weighting can correct the distortion when the inclusion probabilities are known or validly estimated, positive, and defined for the relevant information.

Worked Example: Correcting a Sample With a Different Class Mix

A hypothetical target portfolio contains 80 claims in class A and 20 in class B. An audit includes 20 class A claims with inclusion probability 0.25 and 16 class B claims with inclusion probability 0.80. Suppose the sample mean severity is exactly $1,000 in class A and exactly $3,000 in class B, and selection within each class is independent of severity. Treat those two observed class means as representative of their respective class populations. Estimate the portfolio mean and compare it with the unweighted audit mean. Use exact claim amounts in dollars; no parametric severity distribution, interest assumption, or mortality assumption is used.

Step 1: Recover the class counts. The weighted class A count is $20/0.25=80$. The weighted class B count is $16/0.80=20$. These reproduce the stated target mix.

Step 2: Standardize the class means to the target mix. The class A severity total is $80(\$1{,}000)=\$80{,}000$. The class B severity total is $20(\$3{,}000)=\$60{,}000$. The target portfolio mean is

\frac{80{,}000+60{,}000}{80+20} =\$1{,}400.00.

Step 3: Compare with the unweighted audit mean. The audit’s unweighted severity total is $20(\$1{,}000)+16(\$3{,}000)=\$68{,}000$, over $20+16=36$ claims. Its mean is $\$68{,}000/36=\$1{,}888.89$, rounded to the nearest cent. The audit overrepresents class B, which has the higher mean severity.

Check. The target weights are $80/100=0.8$ for class A and $20/100=0.2$ for class B. Thus $0.8(\$1{,}000)+0.2(\$3{,}000)=\$1{,}400.00$, as above. The common mistake avoided is assuming that a correct within-class severity estimate automatically produces a correct portfolio-wide estimate.

Practical Conditions and Common Exam Traps

Before applying weights, establish which population the analysis is meant to represent and why claims were selected. The required weighting depends on the design. If the selection probability varies by claim, use claim-level probabilities; if claims are sampled in known strata, the stratum’s inclusion probability may apply to each claim in that stratum. For a weighted mean, the weights enter both numerator and denominator. For a weighted severity distribution, they enter both the weighted count below a threshold and the total weight.

A frequent error is to use $\pi_i$ rather than $1/\pi_i$. In the second example, multiplying each severity by its inclusion probability and normalizing would give

\frac{60(1{,}000)(0.60)+30(3{,}000)(0.30)+10(9{,}000)(0.10)} {60(0.60)+30(0.30)+10(0.10)} = \frac{72{,}000}{46} = \$1{,}565.22.

That is not the inverse-probability estimate. It gives relatively more influence to claims that were already easier to include, worsening rather than correcting the representation. The correct weights are $1/0.60$, $1/0.30$, and $1/0.10$, which give the estimate $\$4{,}333.33$ calculated above.

Do not confuse sampling with truncation or censoring. Inverse-probability weights address unequal inclusion when the selection process and observed response are appropriately defined. They do not supply an unobserved severity for a censored claim or automatically correct a severity threshold that changes the observed response.
Do not ignore zero inclusion probabilities. If a type of claim could not be selected, its contribution cannot be recovered by assigning a finite inverse-probability weight. The target must be narrowed, the sample design changed, or a separate modeling assumption introduced and defended.
Do not mistake sample size for represented population size. The sum of inverse-probability weights can estimate a population count, as in the examples, but it need not equal the number of sampled claims.
Do not assume weighting resolves every model problem. Incorrect inclusion probabilities, missing predictors that drive selection, an unsuitable severity response, and a misspecified conditional model can still invalidate the result.

Key takeaway: Inverse-probability weighting gives claim $i$ representation proportional to $1/\pi_i$. It can restore a target portfolio severity distribution when selection probabilities are valid and positive, the response matches the target, and the selection design is properly reflected in estimation and inference.

Check Your Understanding

For each problem, use the stated target and selection probabilities. Amounts are hypothetical claim severities in dollars at a common valuation date.

A sample has eight claims of $2,000, each with inclusion probability 0.40, and two claims of $6,000, each with inclusion probability 0.20. Find the normalized inverse-probability estimate of the target mean. (Answer: $3,333.33)
A sample has 40 claims of $1,000 with inclusion probability 0.50 and 20 claims of $5,000 with inclusion probability 0.25. Find the estimated target proportion in each severity group and the target mean. (Answer: 50% in each group; mean $3,000.00)
A target portfolio has classes A and B. The audit contains 12 A claims with inclusion probability 0.30 and 18 B claims with inclusion probability 0.60. The respective representative class means are $2,000 and $4,000. Estimate the portfolio mean. (Answer: $2,857.14)
A sample contains one claim each at severities $1,000, $2,000, and $8,000, with inclusion probabilities 0.50, 0.25, and 0.10. Calculate the normalized inverse-probability mean. (Answer: $5,625.00)
A target population includes claims of $1,000, $5,000, and $10,000, but claims at $10,000 have inclusion probability zero. Can inverse-probability weighting alone recover the full target severity distribution? State why. (Answer: No; the $10,000 claims cannot appear in the sample, so their distributional contribution cannot be recovered with finite inverse-probability weights.)

What You'll Learn