Why the Observation Mechanism Matters
A claim database records what was observed, not necessarily every claim and its full amount. A deductible may prevent smaller losses from entering the file; a policy limit may make large losses appear only at the cap. If either mechanism is mistaken for complete observation, empirical frequencies and severity summaries can describe a different population from the one intended.
The diagnostic question is not merely whether an amount has been altered. It is which claims can enter the data, and how much of each included claim is known? That question determines whether a record is complete, left-truncated, right-censored, or subject to more than one mechanism.
Read the Record Together with the Policy Terms
A policy term alone does not prove how a database was constructed. The analyst must check whether the file contains all claim events, whether it records gross losses or payments, and whether capped amounts are flagged. For example, a deductible can lead to left truncation if below-threshold claims are absent. If all claims are retained but only the insurer’s payment is recorded, the data are not a complete sample of gross loss severities; the recorded payment reflects the contract.
Likewise, an amount equal to a policy limit is a warning sign, not conclusive evidence of censoring by itself. The record may be a payment capped at the limit, a gross loss known only to reach the limit, or an exact loss that happens to equal it. Claim-level flags, gross-loss fields, policy terms, and data documentation help distinguish these cases.
| Mechanism | What is observed? | Main implication |
|---|---|---|
| Complete observation | Included claims and their exact severities | Empirical summaries describe the observed target population if the sample is representative |
| Left truncation | Only claims satisfying an inclusion threshold, such as \(X>d\) | Observed severities represent the selected upper portion, not the full severity distribution |
| Right censoring | Exact values below a threshold; at or above it, only that the value reaches the threshold | Capped amounts do not reveal the size of the upper-tail losses |
What Left Truncation Does to a Severity Sample
Suppose claim severities follow a distribution with cumulative distribution function \(F_X(x)=P(X\le x)\). If the data include a claim only when \(X>d\), the observed severity distribution is conditional on passing that threshold. For \(x>d\), its cumulative distribution function is
This result follows directly from conditional probability: the event \(X\le x\) among included claims is the event \(d<X\le x\), and the probability of inclusion is \(1-F_X(d)\). Thus an empirical distribution made from the retained records estimates the conditional distribution, not \(F_X\) for all claims. In particular, its sample mean estimates the mean severity among claims exceeding \(d\), not the overall mean.
Without additional information or assumptions about the missing claims, the below-threshold severity distribution cannot be reconstructed from the retained severities alone. If the file also omits the number of excluded claims, the recorded count cannot by itself give the total claim frequency. A separate event count or exposure-based reconciliation may help with frequency, but does not supply the missing severity amounts.
Worked Example: A Deductible-Based Reporting Threshold
A hypothetical data extract includes gross claim severities only when they exceed \( \$5{,}000 \). The extract contains \( \$6{,}000,\ \$9{,}000,\ \$12{,}000,\ \$16{,}000,\) and \( \$20{,}000 \). An independent audit confirms that the underlying set of eight claims also included losses of \( \$1{,}000,\ \$3{,}000,\) and \( \$4{,}000 \). Find the mean in the extract and compare it with the mean of all eight claims. Name the observation mechanism.
Method. First calculate the observed-sample mean. Then use the audit benchmark to calculate the full-sample mean. The extract’s inclusion rule is \(X>\$5{,}000\), so it is left-truncated at that threshold.
The sum of the five retained amounts is
The extract mean is \( \$63{,}000/5=\$12{,}600.00\). The three excluded amounts sum to \(1{,}000+3{,}000+4{,}000=\$8{,}000\). Therefore the total for all eight claims is \(63{,}000+8{,}000=\$71{,}000\), and the full-sample mean is \( \$71{,}000/8=\$8{,}875.00\).
The extract mean exceeds the all-claim mean by \(12{,}600-8{,}875=\$3{,}725.00\). The difference occurs because the extract excludes the smaller claims; it is not evidence that the underlying severity distribution changed. Check: the all-claim total is \( \$71{,}000\), and \(8\times \$8{,}875=\$71{,}000\). The common mistake is to report \( \$12{,}600.00\) as the mean of all claims; it is only the mean conditional on exceeding \( \$5{,}000\).
What Right Censoring Does to a Severity Sample
Under right censoring at \(u\), a record at the censoring point says that the claim reached \(u\), not that its true severity equals \(u\). When the database stores \(\min(X,u)\), exact observations below \(u\) remain usable as exact severities, while a capped value represents \(X\ge u\). If losses above the cap are genuinely greater than the cap, treating the capped amounts as exact values pulls the empirical mean downward.
The capped data do not identify the exact upper-tail severities. For example, three claims recorded at \( \$20{,}000\) might have gross severities of \( \$21{,}000,\ \$40{,}000,\) and \( \$200{,}000\); the capped records alone cannot distinguish them. A plain empirical distribution of the stored amounts therefore has a point mass at the limit that reflects the recording rule, not necessarily the true concentration of losses at that exact amount.
Worked Example: Three Claims Recorded at the Limit
A hypothetical portfolio has a gross-loss reporting cap of \( \$20{,}000 \). Five claim records show \( \$8{,}000,\ \$14{,}000,\ \$20{,}000,\ \$20{,}000,\) and \( \$20{,}000 \). Claim notes confirm that each record at the cap represents a gross loss of at least \( \$20{,}000 \), but give no exact amount above it. Calculate the naïve mean of recorded amounts and the lower bound for the true mean gross severity.
Method. Calculate the mean of the stored amounts, then replace each of the three censored records by its known lower bound to obtain the smallest possible total consistent with the data.
The recorded total is \(8{,}000+14{,}000+20{,}000+20{,}000+20{,}000=\$82{,}000\). The naïve recorded mean is \(82{,}000/5=\$16{,}400.00\).
The first two severities are exact and sum to \(8{,}000+14{,}000=\$22{,}000\). The other three are each at least \( \$20{,}000\), so their combined severity is at least \(3(20{,}000)=\$60{,}000\). Thus the true total is at least \(22{,}000+60{,}000=\$82{,}000\), and the true mean is at least \(82{,}000/5=\$16{,}400.00\). Because the notes say “at least,” equality is possible if all three underlying severities equal the cap; if each is known to be strictly greater, the true mean is strictly greater than this bound.
The data do not determine the true mean beyond that bound. Check: the mean of the observed capped amounts is \( \$16{,}400.00\), consistent with the lower bound. The common mistake is to interpret each cap as an exact gross loss and claim that the true mean is known to be \( \$16{,}400.00\).
When Both Mechanisms Occur
A dataset can be left-truncated at one threshold and right-censored at another. For example, a file may omit claims at or below a reporting threshold \(d\), while also recording no more than a policy limit \(u\), with \(d<u\). The file then contains only claims selected by \(X>d\), and claims reaching \(u\) may have unknown exact severities. Its uncapped, retained amounts are not a complete sample of all claims.
Worked Example: Diagnose Counts and Severity Separately
A hypothetical insurer reviews \(350\) policy-years. Its claim register confirms seven claim events in total. A severity extract includes gross losses only when they exceed \( \$5{,}000\), and records no amount above a \( \$25{,}000\) cap. The extract lists \( \$7{,}000,\ \$12{,}000,\ \$19{,}000,\ \$25{,}000,\ \$25{,}000\); the register shows that two other claims were below \( \$5{,}000\). The two capped records are confirmed to be losses of at least \( \$25{,}000\). Diagnose the mechanisms, calculate the claim frequency, and calculate the recorded mean for the five included claims.
Method. Use the register for the event count, the threshold and limit for the severity mechanisms, and the extract only for the mean of the included recorded amounts.
The two losses below \( \$5{,}000\) are absent from the severity extract, so the severity sample is left-truncated at \( \$5{,}000\). The two records at \( \$25{,}000\) are right-censored at the cap because their exact severities are known only to be at least that amount. Both mechanisms are present.
Because the register confirms all seven events, the claim frequency per policy-year is \(7/350=0.02\). Using the five severity records alone would give \(5/350=0.0142857\ldots\), or \(0.0143\) claims per policy-year to four decimals, which misses the two below-threshold claims.
The five recorded amounts sum to \(7{,}000+12{,}000+19{,}000+25{,}000+25{,}000=\$88{,}000\). Their recorded mean is \(88{,}000/5=\$17{,}600.00\). This is a mean of the recorded amounts for claims exceeding the threshold, with the upper two amounts capped; it is neither the mean of all seven gross severities nor a known exact mean of the five included gross severities. Check: the register, not the truncated severity extract, supplies the frequency numerator of seven. The common mistake is to use the severity-file row count as the total event count and treat both capped amounts as exact losses.
A Practical Diagnostic Sequence
Use the following sequence before calculating an empirical severity summary. It keeps claim inclusion, amount recording, and exposure-based counting distinct.
Decide whether the analysis concerns gross loss, insurer payment, or another stated amount. A payment is not automatically the same variable as gross severity.
Ask whether claims below a threshold are absent. If inclusion requires \(X>d\), the retained severity sample is left-truncated; do not treat its empirical distribution as the full distribution.
Look for a policy cap, a pile-up at a maximum, or a censoring flag. If a record means only \(X\ge u\), it is right-censored rather than an exact observation of \(X=u\).
Determine whether the number of claim records equals the number of all events. A complete event register or exposure reconciliation may preserve frequency information even when severity records are truncated.
Label an empirical mean or distribution according to the observed population and recording rule. If unobserved severities cannot be recovered, state that limitation rather than implying the data identify them.
Common Exam Trap: A Record at a Threshold Is Not Automatically Exact
Two different errors often arise from the same file. First, an analyst sees no small claims and assumes the observed severity sample is representative of all claims. If low losses were excluded, the sample is left-truncated and its mean generally reflects the selected claims. Second, an analyst sees several values equal to a policy limit and enters them as exact gross losses. If those values are capped records, their true severities are only known to reach the limit.
The correction is to determine the observation rule before summarizing. A claim exactly equal to a limit may be an exact loss, so the cap alone is not enough to declare censoring. Conversely, a file with no values above the limit may still be censored if the recording system caps them. The policy terms and claim-level evidence must agree with the interpretation.
Check Your Understanding
For each situation, identify the observation mechanism and state what the available records support.
- A database contains every reported claim, including claims below a \( \$2{,}000\) deductible, and records exact gross losses for each one. Is the gross severity sample complete, truncated, or censored? (Answer: Complete observations, assuming all target claims are represented.)
- A research file includes only claims with gross loss greater than \( \$8{,}000\). It records their exact amounts, but has no count of claims below the threshold. What distribution does the empirical severity sample represent, and can its row count alone establish total claim frequency? (Answer: The severity distribution conditional on \(X>\$8{,}000\); no.)
- Four records show \( \$6{,}000,\ \$10{,}000,\ \$15{,}000,\) and \( \$15{,}000\). A documented recording cap of \( \$15{,}000\) confirms the last two losses are at least the cap. What is the recorded mean and the lower bound for the mean true severity? (Answer: Recorded mean \( \$11{,}500.00\); true mean at least \( \$11{,}500.00\).)
- A register has 12 claim events over 600 policy-years. A separate severity file retains only losses above \( \$4{,}000\), and has 9 rows. What frequency follows from the complete register, and what error results from using the severity-file count instead? (Answer: \(12/600=0.0200\) claims per policy-year; using 9 gives \(0.0150\), an underestimate.)
- A policy has a \( \$3{,}000\) reporting threshold and a \( \$30{,}000\) cap. Only claims exceeding the threshold appear; values at the cap are flagged as having unknown exact gross loss. What mechanisms apply, and what does a simple mean of the recorded amounts fail to represent? (Answer: Left truncation at \( \$3{,}000\) and right censoring at \( \$30{,}000\); the simple mean is not the mean of all claim severities and does not identify the exact mean of the included gross losses.)