Why the Observation Mechanism Matters
A severity data set may not reveal every loss amount. A system might report only that a claim exceeded its coverage limit, record a loss only within a range, or omit claims below a reporting threshold. These records do not all provide the same information about the loss distribution. In a likelihood, each record must contribute the density or probability corresponding to what was actually observed.
The central distinction is whether a loss was observed but not measured exactly (censoring), or whether a loss was not included in the data at all because it failed a selection rule (truncation). Confusing the two generally changes the likelihood and can change the resulting parameter estimates.
Classifying Incomplete Severity Records
The record description determines the contribution. The table gives the standard forms for a continuous severity model. Endpoint conventions do not affect these formulas when the distribution is continuous, because a single point has probability zero.
| Record type | What is known | Likelihood contribution |
|---|---|---|
| Exact | The loss is observed as \(x\) | \(f_X(x)\) |
| Left-censored at \(c\) | \(X\le c\) | \(F_X(c)\) |
| Right-censored at \(c\) | \(X>c\) | \(S_X(c)\) |
| Interval-censored | \(a<X\le b\) | \(F_X(b)-F_X(a)\) |
| Left-truncated at \(L\) | Only losses with \(X>L\) enter the data | For an exact observed \(x>L\): \(f_X(x)/S_X(L)\) |
| Right-truncated at \(U\) | Only losses with \(X<U\) enter the data | For an exact observed \(x<U\): \(f_X(x)/F_X(U)\) |
For example, if a claim is known to exceed a policy limit \(c\), the claim was recorded and its amount is right-censored: its contribution is \(S_X(c)\). If claims above \(U\) are never present in the data, the sample is right-truncated; an observed exact loss \(x<U\) then has a conditional-density contribution. The limit may look similar in the two descriptions, but the observation mechanisms are different.
Why Truncation Requires Conditioning
Suppose losses are drawn from \(f_X\), but the analyst sees a loss only when it passes a selection rule. For left truncation at \(L\), inclusion means \(X>L\). Among included losses, the relevant density is the conditional density given inclusion. For \(x>L\), the conditional-density formula gives:
The denominator is needed because the observed sample is drawn from the part of the original distribution that passed the threshold. It normalizes the density over \(x>L\). In contrast, censoring retains a record that conveys an event such as \(X>c\); it contributes the probability of that event, not a conditional density given inclusion.
More generally, if a truncated record establishes an event \(A\) and inclusion requires an event \(B\), its contribution is \(P(X\in A\mid X\in B)\). For example, an interval-censored loss in a left-truncated sample contributes the probability of lying in the recorded interval conditional on exceeding the entry threshold. The interval must be intersected with the region eligible for inclusion.
Building a Likelihood from Record Contributions
If records are independent under the model, the likelihood is the product of their contributions. The relevant likelihood is conditional on the observation mechanism when records are selected by truncation. The following examples use an exponential severity model to make the distinction and arithmetic explicit.
Worked Example: Classifying Claims with Different Observation Rules
A hypothetical insurer models positive claim amounts \(X\) with a continuous severity distribution. Its file contains four entries: an exact amount of $7,400; a report that a claim was below $1,000; a claim known to exceed a $25,000 coverage limit; and a rounded amount of $4,000, where amounts are rounded to the nearest $500. Identify the record type and contribution for each entry.
Exact amount. The loss is observed at \(x=7{,}400\), so the contribution is \(f_X(7{,}400)\). Because the amount is in dollars, the density is per dollar.
Below $1,000. This is left-censored: the claim is represented by the event \(X\le1{,}000\). Its contribution is \(F_X(1{,}000)\), not the density evaluated at $1,000.
Exceeds $25,000. This is right-censored: the record establishes \(X>25{,}000\). Its contribution is \(S_X(25{,}000)\).
Rounded to $4,000. Rounding to the nearest $500 means the underlying amount lies from $3,750 to $4,250, apart from an immaterial endpoint convention for a continuous model. The record is interval-censored, with contribution \(F_X(4{,}250)-F_X(3{,}750)\).
Check. These contributions refer, respectively, to a point measurement, a lower-tail event, an upper-tail event, and an interval event. None of the four entries says that claims outside a threshold were excluded from the data, so none is a truncation contribution.
Worked Example: Likelihood for a Mixed Set of Records
A hypothetical data set contains five independent records from an exponential severity model with rate \(\lambda=0.25\) per thousand dollars. Amounts are measured in thousands of dollars, so \(f_X(x)=0.25e^{-0.25x}\), \(F_X(x)=1-e^{-0.25x}\), and \(S_X(x)=e^{-0.25x}\). The records are: an exact loss \(x=2\); a left-censored loss \(X\le1\); a right-censored loss \(X>3\); an interval-censored loss \(1<X\le2.5\); and an exact loss \(x=2\) from a sample left-truncated at \(L=1\). Find each contribution and their product.
1. Exact loss \(x=2\). The contribution is the density: \(f_X(2)=0.25e^{-0.5}=0.1516327\).
2. Left-censored at 1. The contribution is: \(F_X(1)=1-e^{-0.25}=0.2211992\).
3. Right-censored at 3. The contribution is: \(S_X(3)=e^{-0.75}=0.4723666\).
4. Interval-censored in \((1,2.5]\). Subtract the lower cumulative probability from the upper one: \(F_X(2.5)-F_X(1)=(1-e^{-0.625})-(1-e^{-0.25})=e^{-0.25}-e^{-0.625}=0.7788008-0.5352614=0.2435394\).
5. Exact loss \(x=2\), observed only when \(X>1\). This is a left-truncated exact observation, so use the conditional density: \(f_X(2)/S_X(1)=0.1516327/0.7788008=0.1947002\).
By independence, multiply the five contributions: \(0.1516327(0.2211992)(0.4723666)(0.2435394)(0.1947002)\approx0.0007513\). The product is a likelihood value for the specified model parameter, not a probability that the parameter equals \(0.25\).
Check. The truncated record has a larger conditional density than its unconditioned density because its likelihood is renormalized over \(X>1\). The other four records use their own exact or event-based contributions, without that truncation adjustment.
Worked Example: Separating a Truncated Exact Record from Censored Records
A hypothetical severity model is exponential with rate \(\lambda=0.2\) per thousand dollars. Consider three independent records: an exact loss \(x=5\) from a sample that includes only losses above \(L=2\); a separate right-censored loss known to exceed 5; and an interval-censored loss \(2<X\le5\) from the left-truncated sample. Compute the contributions and joint likelihood.
Exact loss from the truncated sample. The conditional density is: \(f_X(5)/S_X(2)=\{0.2e^{-0.2(5)}\}/e^{-0.2(2)}=0.2e^{-0.6}=0.1097623\).
Right-censored record. This record is not described as having been selected by a threshold; it simply establishes \(X>5\). Its contribution is: \(S_X(5)=e^{-1}=0.3678794\).
Interval-censored record in the truncated sample. Given inclusion \(X>2\), the record establishes \(2<X\le5\). Its conditional probability is: \(\{F_X(5)-F_X(2)\}/S_X(2)=\{S_X(2)-S_X(5)\}/S_X(2)=1-e^{-0.6}=0.4511884\).
Joint likelihood. Multiplying gives \(0.1097623(0.3678794)(0.4511884)\approx0.01822\). This calculation conditions only the records from the truncated sample. The separate right-censored observation contributes its ordinary survival probability.
Common Likelihood Errors
- Using a density for a censored record. A statement such as \(X>c\) gives an event, not an exact value. Use \(S_X(c)\), not \(f_X(c)\).
- Treating truncation as censoring. If losses below \(L\) are absent, the exact observed values come from a conditional distribution. Use \(f_X(x)/S_X(L)\) for left truncation, not just \(f_X(x)\).
- Reversing a tail probability. Left-censoring at \(c\) gives \(F_X(c)\); right-censoring at \(c\) gives \(S_X(c)=1-F_X(c)\).
- Forgetting the interval subtraction. An interval \((a,b]\) has probability \(F_X(b)-F_X(a)\), not \(F_X(b)\) alone.
- Applying a truncation adjustment to every record. Condition only on the selection rule that actually governed that record’s inclusion.
These formulas assume that the thresholds and observation rules are known and that the censoring or selection mechanism does not add unmodeled information about severity. If the mechanism depends on the loss in a way not captured by the stated rule, the likelihood must account for that mechanism rather than treating the threshold as fixed and noninformative.
Check Your Understanding
For each problem, identify the observation mechanism and write or evaluate the corresponding contribution. Amounts are in thousands of dollars where stated.
- A continuous severity model records an exact loss of 4.2, a loss known to be at most 1, a loss known to exceed 10, and a loss known to lie in \((6,8]\). Write the four contributions in order. (Answer: \(f_X(4.2),\ F_X(1),\ S_X(10),\ F_X(8)-F_X(6)\).)
- For an exponential severity with rate \(\lambda=0.1\) per thousand dollars, a record is left-censored at 3. Compute its contribution to four decimal places. (Answer: \(F_X(3)=1-e^{-0.3}=0.2592\).)
- For an exponential severity with rate \(\lambda=0.5\) per thousand dollars, an exact loss \(x=4\) is observed only when \(X>1\). Find the conditional-density contribution. (Answer: \(f_X(4)/S_X(1)=0.5e^{-1.5}\approx0.1116\) per thousand dollars.)
- A Weibull severity has density \(f_X\) and distribution function \(F_X\). Only losses below \(U=8\) enter the data, and an included loss is observed exactly as \(x=5\). Write its contribution. (Answer: \(f_X(5)/F_X(8)\).)
- An exponential severity has rate \(\lambda=0.2\) per thousand dollars. A loss is included only if \(X>1\), and its record says \(2<X\le4\). Compute its conditional probability to four decimal places. (Answer: \(\{F_X(4)-F_X(2)\}/S_X(1)=e^{-0.2}-e^{-0.6}\approx0.2699\).)