Why the Recording Mechanism Matters
A loss database does not necessarily contain every loss at its exact amount. A reporting rule may exclude some losses from the database altogether, or it may retain a claim while recording only that its amount falls above or below a threshold. These mechanisms create different data, even when they use the same dollar threshold.
The distinction is fundamental: truncation determines whether a loss appears in the data at all, while censoring determines how precisely an included loss is measured. A table of exact recorded amounts alone may not reveal which mechanism operated. The record-generation process must also be understood.
For continuous loss models, using a strict or non-strict inequality at a fixed threshold does not change the probabilities, since the probability of an exact value at that threshold is zero. In actual records, the convention for a loss exactly equal to a threshold should still be specified.
Four Mechanisms and What the Record Contains
For a loss with distribution function \(F_X(x)=P(X\leq x)\) and survival function \(S_X(x)=P(X>x)\), the four one-sided mechanisms can be described by what qualifies for observation and what the record says.
| Mechanism | Which losses appear? | What is recorded? | What is missing? |
|---|---|---|---|
| Left truncation at \(L\) | Losses above \(L\) | Exact loss for each included record | Losses at or below \(L\) have no record |
| Right truncation at \(U\) | Losses below \(U\) | Exact loss for each included record | Losses at or above \(U\) have no record |
| Left censoring at \(C\) | All losses, including those at or below \(C\) | For a low loss, a statement such as \(X\leq C\); otherwise, the exact loss | The exact amount of a low loss |
| Right censoring at \(C\) | All losses, including those at or above \(C\) | For a high loss, a statement such as \(X>C\); otherwise, the exact loss | The exact amount of a high loss |
Thus, left and right describe the side of the loss scale where information is restricted. They do not, by themselves, tell you whether a record is absent or merely imprecise. The word truncation signals absence; censoring signals an included observation with incomplete amount information.
A common source of censoring in insurance records is a coverage limit. If the data system retains each claim and marks claims reaching the limit as “at least the limit,” the underlying loss is right-censored at that amount. If instead the database contains only claims below the limit and has no record of higher claims, the observed sample is right-truncated. The payment made by an insurer is not automatically the underlying loss: a capped payment may be an exact payment amount while still providing only a lower bound on the loss.
The same distinction applies on the low side. If all claims are retained but claims below a reporting threshold are marked “at most the threshold,” the data are left-censored. If claims below that threshold are not entered into the data at all, the sample is left-truncated.
Probability Calculations Behind the Labels
The probability calculations follow from partitioning the loss scale at the threshold. For example, at threshold \(C\), every loss is either at or below \(C\), or above \(C\). The probabilities of those disjoint events sum to one:
This gives the relevant percentages, but not the mechanism itself. The mechanism comes from the record-keeping rule: an excluded value creates truncation, whereas a retained record stating only a bound creates censoring.
For censoring, the corresponding probabilities describe the kind of record, not whether a loss appears. Left-censored records occur with probability \(F_X(C)\); right-censored records occur with probability \(S_X(C)\). In either case, those losses are included in the data with a bound rather than an exact amount.
This distinction also clarifies what can be counted. If the total number of losses under review is known, censored losses can be counted as included records. Under truncation, the omitted losses do not appear in the observed sample; the total number of losses that generated the sample may not be available from that sample alone.
Worked Example: A Left-Truncated Reporting Threshold
A hypothetical insurer studies ground-movement claims with positive loss amount \(X\). Its extract includes only claims exceeding $500, and included claims have their exact amounts recorded. Assume \(X\) has an exponential distribution with mean $1,000, so its rate is \(1/1{,}000\) per dollar. The insurer initially handled 2,000 claims. Find the probability a claim is represented in the extract and the expected number of included claims. No interest or mortality assumptions apply to this loss-record problem.
Method. Because claims at or below $500 are not in the extract, this is left truncation at \(L=\$500\), not left censoring. For an exponential loss with mean \(\theta=\$1{,}000\), \(S_X(x)=e^{-x/\theta}\).
The probability of being included is
The omitted proportion is the complement:
The expected number included among 2,000 claims is
The check is that the expected omitted count is \(2{,}000(0.3934693\ldots)\approx786.9\), and \(1{,}213.1+786.9=2{,}000.0\). The probabilities also sum to \(0.6065+0.3935=1.0000\). Counts are expected values, not a claim that the actual extract must contain a fractional number of claims. The common mistake avoided here is describing the below-threshold claims as censored: they have no records at all.
Worked Example: Right-Censored Claims at a Coverage Limit
A hypothetical data file retains every claim from a portfolio of 1,000 claims. Claims below $5,000 have exact loss amounts; claims at or above $5,000 are flagged “at least $5,000.” Assume the underlying loss \(X\) is exponential with mean $2,000, with rate \(1/2{,}000\) per dollar. Find the probability of a right-censored record and the expected numbers of exact and censored records. No interest or mortality assumptions apply.
Method. The high claims are retained with a lower bound on their amounts, so the mechanism is right censoring at \(C=\$5{,}000\). The probability of a censored record is the survival probability:
The probability of an exact record below the limit is
Therefore, the expected number of censored records is \(1{,}000(0.0820850\ldots)\approx82.1\), and the expected number of exact records is \(1{,}000(0.9179150\ldots)\approx917.9\). Their sum is \(1{,}000.0\), and their proportions sum to \(1.0000\).
These are expected counts; the database contains all 1,000 claims regardless of how many are censored. The common mistake avoided is calling claims at the cap right-truncated merely because their exact losses are unknown. The “at least $5,000” flags establish that those claims remain in the records.
Worked Example: Right Truncation Versus Right Censoring
Consider a hypothetical set of 400 losses with \(X\) exponentially distributed with mean $1,500, rate \(1/1{,}500\) per dollar. Compare two possible data systems using a $3,000 threshold: System A records exact losses only when they are below $3,000 and omits higher losses entirely. System B retains all 400 claims but records each loss of $3,000 or more as “at least $3,000.” Calculate the inclusion or censoring probabilities and corresponding expected counts. No interest or mortality assumptions apply.
Method. System A applies right truncation. Its inclusion probability is the probability of a loss below the threshold:
The omitted probability is \(e^{-2}=0.1353\). Thus, the expected number in System A’s extract is \(400(0.8646647\ldots)\approx345.9\), and the expected number omitted is \(400(0.1353353\ldots)\approx54.1\). These expected counts sum to \(400.0\).
System B applies right censoring. The probability of a censored record is \(P(X\geq3{,}000)=e^{-2}=0.1353\), giving an expected \(400(0.1353353\ldots)\approx54.1\) censored records. The expected number of exact records below the threshold is \(400(0.8646647\ldots)\approx345.9\). In System B those two types of records sum to all 400 claims; in System A only the below-threshold records appear.
The matching expected counts by threshold do not make the data mechanisms identical. The decisive check is whether the higher claims are absent or present with a bound. The common mistake avoided is inferring truncation just from seeing no exact amounts above $3,000; System B explicitly retains those claims as censored records.
Worked Example: Left-Censored Small Claims
A hypothetical administrator retains all 800 claims in a file. Claims of $250 or less are labeled “at most $250”; larger claims have exact amounts. Assume \(X\) is exponential with mean $500, rate \(1/500\) per dollar. Find the probability of a left-censored record and the expected numbers of left-censored and exact records. No interest or mortality assumptions apply.
Method. The small claims remain represented, but their exact amounts are replaced by an upper bound. This is left censoring at \(C=\$250\). The probability of a left-censored record is
The probability of an exact record is \(P(X>250)=e^{-0.5}=0.6065\). The expected number of left-censored records is \(800(0.3934693\ldots)\approx314.8\); the expected number of exact records is \(800(0.6065307\ldots)\approx485.2\). The expected counts sum to \(800.0\), as required because every claim is retained.
The common mistake avoided is calling the small claims left-truncated because their exact values are unavailable. The “at most $250” entries show that they are in the file; their amounts are censored, not omitted.
Practical Checks and Common Traps
When examining a loss extract, identify the record rule before interpreting the values. In particular, ask whether every relevant claim has a row, whether each row gives an exact underlying loss or only a bound, and whether a displayed amount is the loss itself or an insurer payment subject to coverage terms.
If losses on one side of a threshold are excluded entirely, classify the observed sample as truncated on that side.
If claims remain in the data with statements such as “at least” or “at most,” classify the affected amounts as censored.
Determine whether the record concerns the underlying loss \(X\), a payment, or another quantity; a payment cap alone does not establish the censoring rule for \(X\).
For left-side mechanisms use the probability below the threshold; for right-side mechanisms use the probability above it. Then decide whether that probability describes omitted or retained records.
A frequent exam trap is to calculate the right probability but attach it to the wrong mechanism. For example, \(S_X(C)\) is the probability of a loss above \(C\). It is the omitted proportion under right truncation if high losses are excluded, but it is the censored-record proportion under right censoring if high losses are retained with a lower bound. The arithmetic may be identical; the interpretation depends on the record rule.
Check Your Understanding
For each hypothetical scenario, identify the mechanism and calculate the requested quantity. Probabilities are rounded to four decimals; expected counts are rounded to one decimal.
- A loss file includes only claims exceeding $200. Losses are exponential with mean $400, and 500 claims were initially handled. What fraction is included, and what is the expected number in the file? (Answer: Left truncation; included fraction \(e^{-0.5}=0.6065\); expected included count \(303.3\).)
- All 300 claims remain in a file, but amounts of $100 or less are recorded as “at most $100.” Losses are exponential with mean $200. What is the probability of a censored record and the expected number of such records? (Answer: Left censoring; \(1-e^{-0.5}=0.3935\); expected count \(118.0\).)
- A hypothetical register contains only losses below $80,000; the loss distribution is uniform from $0 to $100,000, and 500 losses were initially in scope. Find the inclusion probability and expected number recorded. (Answer: Right truncation; inclusion probability \(0.8000\); expected count \(400.0\).)
- All 250 claims are retained. Claims of at least $4,000 are marked “at least $4,000”; losses are exponential with mean $2,000. Find the censored probability and expected number censored. (Answer: Right censoring; \(e^{-2}=0.1353\); expected count \(33.8\).)
- Two files use a $1,000 threshold for high losses. File A has no rows for losses above $1,000. File B retains those claims and labels them “greater than $1,000.” Classify each file and state the difference in what its high-loss records contain. (Answer: File A is right-truncated; File B is right-censored. A has no high-loss records, while B has records with a lower bound but no exact amounts.)