Interpreting Fits Under Truncation and Censoring

Why the Interpretation of a Severity Fit Depends on the Data

A severity model is meaningful only in relation to the amount being modeled and the claims that could enter the data. A deductible may exclude losses below a threshold, while a policy limit may prevent the full loss from being observed. Fitting a model to the recorded amounts without accounting for either mechanism can change the quantity being estimated.

The central distinction is between ground-up severity, the full loss before coverage adjustments; conditional severity, the loss distribution among claims selected by a threshold; and limited-payment severity, the amount payable under a specified coverage limit. A fit may accurately describe one of these targets without estimating the others.

Definition: Let $X$ denote a ground-up loss. If a claim is observed only when $X>d$, the data are left-truncated at $d$: losses at or below $d$ are absent, not recorded as zero. If a claim is observed but its amount is known only to exceed a limit $u$, it is right-censored at $u$. If the modeled response is $W=\min(X,u)$, the response is a limited payment; values at $u$ represent amounts that may exceed the limit, not necessarily losses equal to $u$.

These mechanisms have different implications. Truncation changes which claims are sampled. Censoring preserves information that a loss reached at least a threshold, even though its full amount is unknown. A limited-payment target deliberately concerns the capped amount. Earlier material on defining severity targets and censoring likelihoods provides the relevant data structures; here the emphasis is on interpreting what a fitted result means.

What a Fit to Deductible-Selected Claims Estimates

Suppose claims enter the data only if $X>d$. If an ordinary severity model is fitted to the selected claims as though they were a random sample from all ground-up losses, its interpretation is generally conditional: it describes the distribution of $X$ given $X>d$, provided the model and fitting procedure are otherwise appropriate. It does not, by selection alone, estimate the unconditional distribution of $X$.

For instance, the conditional mean is

E[X\mid X>d]=\frac{E[X\,\mathbf{1}(X>d)]}{P(X>d)}.

This is not generally $E[X]$. To infer ground-up severity from truncated data, the analyst needs a model for the ground-up distribution and a likelihood or other method that accounts for selection. The relationship between conditional and ground-up quantities depends on that distribution and on the threshold.

Worked Example: Interpreting an Exponential Fit Above a Deductible

In a hypothetical portfolio, only losses above $d=\$500$ are included. The three observed ground-up losses are $\$900$, $\$1{,}300$, and $\$2{,}100$. Assume independent ground-up losses follow an exponential distribution with mean parameter $\theta$, so $f(x)=\theta^{-1}e^{-x/\theta}$ for $x>0$. Estimate $\theta$ using the likelihood conditional on selection, then state the fitted conditional gross mean.

Method and assumptions. The exponential model is specified for ground-up loss, claims are selected exactly when $X>d$, and the three observed losses are exact. For an exponential distribution, $P(X>d)=e^{-d/\theta}$. Thus an exact selected loss has conditional density $f(x)/P(X>d)$ for $x>d$.

Step 1: Calculate excesses above the threshold. They are $\$900-\$500=\$400$, $\$1{,}300-\$500=\$800$, and $\$2{,}100-\$500=\$1{,}600$. Their total is $\$2{,}800$.

Step 2: Estimate the exponential mean. Under the conditional exponential model, each excess $X-d$ has an exponential distribution with mean $\theta$. The maximum-likelihood estimate is the average excess:

\hat{\theta}=\frac{400+800+1{,}600}{3} =\frac{2{,}800}{3} =\$933.33.

Step 3: Interpret the result. This estimates the ground-up exponential mean because the exponential model’s memoryless property makes the excess distribution above a fixed threshold exponential with the same mean. The fitted conditional gross mean is $d+\hat{\theta}=\$500+\$933.33=\$1{,}433.33$. These are different quantities.

Check and trap avoided. The average of the selected gross losses is also $(900+1{,}300+2{,}100)/3=\$1{,}433.33$, agreeing with the fitted conditional gross mean. Treating that average as the ground-up mean would be wrong: selection removed all losses at or below $\$500$. This conversion relies on the exponential assumption and fixed threshold; it is not a general property of severity distributions.

The exponential case is especially convenient, but it should not be generalized without justification. For many distributions, $E[X\mid X>d]$ depends on both the ground-up parameters and $d$, and the conditional distribution may not even belong to the same simple model family. A conditional fit can still support conclusions about the selected population, such as comparing observed conditional severities across groups, if selection and model specification are handled consistently. It does not by itself support a conclusion about all ground-up losses.

What Censoring Contributes to a Fit

When a loss reaches a policy limit, its full value is unknown, but the observation is not uninformative. A claim censored at $u$ tells us $X\ge u$. For a model with density $f_\theta$ and survival function $S_\theta(x)=P(X>x)$, an exact observation contributes its density to the likelihood, while a right-censored observation contributes its survival probability. When the data are also selected above a deductible $d$, these contributions must be conditional on $X>d$.

L(\theta) = \prod_{i\in E}\frac{f_\theta(x_i)}{S_\theta(d_i)} \prod_{i\in C}\frac{S_\theta(c_i)}{S_\theta(d_i)}.

Here $E$ indexes exact claims, $C$ indexes censored claims, $d_i$ is the selection threshold, and $c_i$ is the censoring point. This likelihood assumes the stated ground-up model is appropriate and censoring is noninformative given modeled covariates: the censoring mechanism does not depend on the unobserved amount in a way omitted from the model. If all claims are observed from ground-up, take $d_i=0$, so $S_\theta(0)=1$ for a positive loss distribution.

Worked Example: Deductible Selection With a Censored Claim

A hypothetical dataset contains claims selected above $d=\$300$. Two exact ground-up losses are $\$700$ and $\$1{,}100$, and a third claim is right-censored at $u=\$1{,}800$. Assume independent losses follow an exponential distribution with mean $\theta$, and censoring at the limit is noninformative. Find the conditional-likelihood estimate of the ground-up mean.

Method. For the exponential model, conditional on $X>d$, the excess $X-d$ is exponential with mean $\theta$. Each exact claim contributes one exponential density for its excess. A censored claim contributes the probability that its excess is at least $u-d$; it contributes exposure to the likelihood but does not count as an exact claim.

Step 1: Compute excess exposures. The first exact excess is $\$700-\$300=\$400$. The second is $\$1{,}100-\$300=\$800$. The censored claim contributes $\$1{,}800-\$300=\$1{,}500$. Total excess exposure is $\$400+\$800+\$1{,}500=\$2{,}700$.

Step 2: Maximize the likelihood. If there are $r$ exact claims and total excess exposure $T$, the exponential likelihood is proportional to $\theta^{-r}e^{-T/\theta}$. Its log-likelihood is $-r\log\theta-T/\theta$, whose derivative is $-r/\theta+T/\theta^2$. Setting this equal to zero gives $\hat{\theta}=T/r$. Here $r=2$, so:

\hat{\theta}=\frac{\$2{,}700}{2}=\$1{,}350.

Check and trap avoided. The estimate is $\$1{,}350$, not $\$2{,}700/3=\$900$: the censored claim contributes exposure, but it does not contribute an exact-event density or an additional numerator count. Ignoring its exposure would also be wrong; that would give $(400+800)/2=\$600$. This estimate is a ground-up mean under the assumed exponential model and fixed truncation threshold, not a general censoring shortcut.

The reason the likelihood retains information from censored claims is visible in the exponential derivation: an exact claim contributes a factor involving the event rate, while a censored claim contributes only the probability of surviving to its censoring point. The latter increases total exposure $T$ without increasing the count $r$ of exact observations. In other severity families, the same exact-density/survival-probability principle applies, but the resulting parameter estimate need not have this simple ratio form.

Limited Payments Are a Different Target

Sometimes the objective is not to recover ground-up severity. An insurer may need the expected amount paid under a policy limit. If the payment is $W=\min(X,u)$, then $E[W]$ is the expected limited payment, not $E[X]$. Modeling the capped amounts directly can be appropriate when this is the intended pricing target, but it should not be interpreted as a ground-up fit.

Worked Example: Converting a Ground-Up Estimate to Expected Limited Payment

Suppose a separate, censor-aware analysis for a hypothetical coverage has produced a ground-up exponential mean estimate of $\hat{\theta}=\$1{,}000$. The policy pays up to a gross-loss limit of $u=\$1{,}000$, so payment is $W=\min(X,u)$. Calculate the expected limited payment under the fitted model.

Method and assumptions. Assume $X$ follows an exponential distribution with mean $\hat{\theta}=\$1{,}000$, and the limit applies to the gross loss with no deductible. For a nonnegative loss, the expectation of a capped amount is the integral of its probability of exceeding each payment level:

E[\min(X,u)]=\int_0^u P(X>t)\,dt.

Step 1: Substitute the exponential survival function. Here $S_X(t)=e^{-t/1000}$, with amounts in dollars. Therefore:

E[W]=\int_0^{1000}e^{-t/1000}\,dt =1000(1-e^{-1000/1000}).

Step 2: Evaluate. Since $e^{-1}\approx0.367879$, $1-e^{-1}\approx0.632121$. Thus $E[W]=1000(0.632121)=\$632.12$, rounded to the nearest cent.

Check and trap avoided. The expected ground-up loss remains $\$1{,}000$, while the expected capped payment is $\$632.12$. The capped expectation must be no greater than both the limit and the ground-up mean, and it is. Substituting the limit as though every claim’s loss were exactly $\$1{,}000$ would overstate expected payment.

Which Conclusions Remain Valid?

A fit affected by truncation or censoring is not automatically useless. Its conclusions are valid when they match the target and the assumptions used to obtain them. The key is to state the population, response, and observation mechanism precisely.

Data and method	Conclusion that may be supported	Conclusion not supported without more
Ordinary fit to exact claims selected above a deductible	Features of the selected, conditional severity distribution, if the fit is appropriate	The ground-up distribution or mean merely from treating selected claims as an unselected sample
Ground-up model fitted with a truncation-adjusted likelihood	Ground-up parameters under the assumed distribution and selection mechanism	Unconditional conclusions if the model or selection mechanism is misspecified
Ground-up model fitted with an appropriate censoring likelihood	Ground-up parameters, using both exact and censored information under stated assumptions	The claim that a censored amount equals its censoring limit
Model of capped payment $W=\min(X,u)$	Expected payment and other properties of that capped response	Ground-up severity without an additional model connecting $W$ to $X$

Comparisons across rating groups need particular care. If groups have different deductibles, limits, or selection rules, differences in fitted conditional severity may reflect different observation conditions as well as differences in ground-up risk. A valid comparison requires comparable targets or an explicit model that accounts for those differences. Similarly, good fit to observed paid amounts does not establish that the ground-up distribution is well calibrated.

Common Interpretation Errors

Calling a selected-sample mean a ground-up mean. The average of losses above a deductible describes the selected observations, unless an appropriate model justifies a conversion.
Treating a limit observation as exact. A claim censored at $u$ contributes information that $X\ge u$, not evidence that $X=u$.
Confusing a limited payment with a censored ground-up loss. They can share the same recorded number, but the first is a coverage-adjusted target and the second is incomplete information about $X$.
Assuming a special distributional result is universal. Exponential memorylessness makes excess losses above a fixed threshold especially simple; other distributions generally require their own conditional likelihood and interpretation.
Reporting a parameter without naming its target. State whether the estimate concerns ground-up loss, loss conditional on selection, excess over a deductible, or a limited payment.

Key takeaway: Truncation changes which claims are represented, censoring records incomplete information about claim size, and a limited-payment response changes the target itself. A fit supports ground-up conclusions only when its model and likelihood account for the observation process; otherwise, describe it as a conditional or coverage-limited fit.

Check Your Understanding

For each problem, identify the target before interpreting the numerical result.

Assume ground-up losses are exponential with mean $\theta$. Claims are observed only above $d=\$200$, and the exact losses are $\$500$, $\$800$, and $\$1{,}100$. Find the conditional-likelihood estimate of $\theta$ and the fitted conditional gross mean. (Answer: $\hat{\theta}=(300+600+900)/3=\$600$; conditional gross mean $=\$200+\$600=\$800$.)
Assume exponential ground-up losses with mean $\theta$, selection above $d=\$100$, and noninformative censoring. Exact losses are $\$400$ and $\$700$; a third claim is censored at $\$900$. Find the estimate of $\theta$. (Answer: excess exposure $=300+600+800=\$1{,}700$; divide by 2 exact claims, giving $\hat{\theta}=\$850$.)
Assume exponential ground-up losses, with two exact observations of $\$250$ and $\$750$, and one observation right-censored at $\$1{,}000$. There is no deductible. Find the censor-aware maximum-likelihood estimate of the mean. (Answer: total exposure $=250+750+1{,}000=\$2{,}000$; divide by 2 exact claims, giving $\hat{\theta}=\$1{,}000$.)
A severity model is fitted to exact losses that enter the dataset only when they exceed a deductible. The fitted distribution describes the selected observations well. What further information is needed before treating its mean as a ground-up mean? (Answer: A specified ground-up distribution and a method that accounts for selection at the deductible; a good conditional fit alone does not establish the ground-up mean.)
Assume $X$ is exponential with ground-up mean $\$800$, and the policy pays $W=\min(X,\$400)$. Calculate the expected payment, rounded to the nearest cent. (Answer: $800(1-e^{-400/800})=800(1-e^{-0.5})=\$314.78$, the expected limited payment, not the ground-up mean.)

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