Introduction
A loss file may contain exact amounts for some claims, only a threshold statement for others, and no record at all for losses below an entry cutoff. These are different observation mechanisms. Treating them as interchangeable can change both the likelihood and the resulting estimate of the severity distribution.
This tutorial combines those mechanisms in a single analysis. The central distinction is that censoring records partial information about a loss that entered observation, while truncation excludes a loss from the observed sample altogether. The methods here are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.
Likelihood Contributions for Mixed Records
Suppose the analysis conditions on inclusion in the data set, and a record has a known truncation threshold \(a\). For an exact observed loss \(x>a\), its likelihood contribution is the density of \(X\) conditional on exceeding \(a\). For a right-censored record at \(c\ge a\), the contribution is the conditional probability that the loss exceeds \(c\). The censoring threshold is assumed known, and the censoring mechanism is assumed independent of the loss, or otherwise noninformative for the severity model.
For independent records, multiply the relevant contribution for each record. If truncation thresholds differ between records, use the threshold applicable to each record. In a likelihood conditional on the observed sample size and inclusion, the truncation denominator is not optional: it accounts for the fact that only losses above the entry threshold could appear.
Record the truncation threshold \(a_i\) for each loss and confirm that an observed loss must exceed it.
Use a density for an exact loss, or a survival probability for a right-censored loss.
Divide each contribution by \(S_X(a_i)\), the probability that the loss would enter the sample.
Multiply the conditional contributions under independent sampling of losses.
Worked Example: Classifying and Writing Three Contributions
A hypothetical file contains three independent claims. The first claim was included only if it exceeded 2 and is reported exactly as 3. The second was included only if it exceeded 2 and is reported as greater than 4. The third was included only if it exceeded 1 and is reported exactly as 5. Write the likelihood contribution for each record and the combined likelihood using a general continuous severity model.
Method. Apply the conditional density to each exact record and the conditional survival probability to the threshold-censored record.
For the first claim, the entry threshold is \(a_1=2\), and its exact loss is \(x_1=3\). Its contribution is \(\frac{f_X(3)}{S_X(2)}\).
For the second claim, the entry threshold is \(a_2=2\), and the record says \(X>4\). Its contribution is \(\frac{S_X(4)}{S_X(2)}\).
For the third claim, the entry threshold is \(a_3=1\), and its exact loss is \(x_3=5\). Its contribution is \(\frac{f_X(5)}{S_X(1)}\).
The claims are independent, so multiply:
The first and third observations are exact, the second is right-censored, and all three are left-truncated. The common mistake avoided here is to treat the censored claim as if its loss were exactly 4; the record gives \(X>4\), so its numerator is \(S_X(4)\), not \(f_X(4)\).
Why Truncation Requires Conditioning
The denominator in each contribution follows directly from conditional probability. Before selection, the density of a loss at \(x\) is \(f_X(x)\). Given that the record is observed, it is already known that \(X>a\). Therefore, for \(x>a\), the conditional density is the joint density of observing a loss near \(x\) and satisfying the entry rule, divided by the probability of satisfying that rule.
Similarly, a censored record at \(c\ge a\) says \(X>c\), an event contained within \(X>a\). Thus \(P(X>c\mid X>a)=S_X(c)/S_X(a)\). The same conditioning principle applies to interval-censored records: if the observed loss is known to lie in \((l,r]\) and the interval is compatible with entry, the conditional contribution is \([F_X(r)-F_X(l)]/S_X(a)\). The interval endpoints and entry threshold must be interpreted consistently.
These expressions assume that the analysis conditions on inclusion. If the number of losses screened out by the truncation rule is also observed and modeled, the likelihood must include information about those nonselected losses; the conditional likelihood above does not use that extra information.
Worked Example: Evaluating a Mixed Likelihood Under an Exponential Model
Assume losses have an exponential distribution with rate \(\lambda=0.5\) per loss unit, so \(f_X(x)=0.5e^{-0.5x}\) and \(S_X(x)=e^{-0.5x}\) for \(x\ge0\). Use the three records and entry thresholds from the preceding example. Calculate the conditional likelihood, rounding the final likelihood to six decimal places.
Method. Substitute the specified density and survival function into each conditional contribution. The rate and units are fixed; this example evaluates a likelihood and does not estimate the rate.
For the exact loss 3 with entry threshold 2:
For the right-censored loss \(X>4\) with entry threshold 2:
For the exact loss 5 with entry threshold 1:
Multiplying the unrounded expressions gives \(L=(0.5e^{-0.5})(e^{-1})(0.5e^{-2})=0.25e^{-3.5}\approx0.007549\). The likelihood is therefore 0.007549 to six decimal places. As a check, each factor is positive and no greater than 1 for these records and parameters. The common mistake avoided is omitting a truncation denominator, which would instead assign likelihood to an unselected, unconditional sample.
Adjusted Empirical Survival With Delayed Entry
A model-free estimate can also use exact event times and right-censoring indicators while adjusting the risk set for left truncation. A record enters the risk set only after its entry time and leaves at its event or censoring time. At each event time, divide the number of events by the number at risk immediately before that time.
The adjustment is in the risk set, not in the event indicator: a loss that had not yet crossed its entry threshold could not have been observed and should not be counted as at risk. The product-limit formula builds survival from successive conditional probabilities of surviving each event time. It estimates the underlying survival curve only under the stated independent-truncation and noninformative-censoring assumptions.
Worked Example: Product-Limit Estimate With Entry and Censoring
A hypothetical severity study records each loss's entry time \(a\), followed by an exact event time or a right-censoring time. The records are \((a,t,\text{status})=(1,2,\text{event}),(1,3,\text{censored}),(2,4,\text{event}),(3,4,\text{censored}),(2,5,\text{event})\). Assume independent truncation and noninformative censoring. Calculate the estimated survival just after time 4.
Method. Form the delayed-entry risk set immediately before each event time at or before 4, then multiply the survival factors. At time 2, the first two records have entered and neither has left, so \(R_2=2\); one event occurs, so \(d_2=1\). The factor is \(1-1/2=1/2\).
At time 4, the first record had an event at 2 and the second was censored at 3, so neither remains at risk. The third, fourth, and fifth records have entered by time 4 and have not left before time 4. Thus \(R_4=3\). One of these has an event at 4, so \(d_4=1\); the other record observed at time 4 is censored and contributes to the risk set immediately before 4, but not to \(d_4\). The second factor is \(1-1/3=2/3\).
The estimated survival just after time 4 is 0.3333, to four decimal places. As a check, it is the product of the estimated survival factors at event times 2 and 4; the censoring at time 3 creates no event factor. The common mistake avoided is counting a not-yet-entered loss in an earlier risk set, or treating a censored record as an event.
Empirical Adjustment When Only a Truncated CDF Is Available
For a fixed truncation point \(a\), the observed losses follow the conditional distribution above \(a\), not the original distribution. For \(t>a\), the conditional CDF is \(F_X(t\mid X>a)=[F_X(t)-F_X(a)]/S_X(a)\). Consequently, if \(F_X(a)\) is supplied by a justified external source or model and the conditional CDF above \(a\) is estimated empirically, the original CDF can be reconstructed above \(a\). Without information about \(F_X(a)\), the truncated observations alone cannot identify how much probability lies at or below \(a\).
Worked Example: Adjusting an Empirical CDF Above a Cutoff
A hypothetical portfolio records only losses exceeding \(a=2\). A justified reference model specifies \(F_X(2)=0.20\), hence \(S_X(2)=0.80\). Among five observed losses, three are at most 4. Estimate \(F_X(4)\) using the empirical conditional CDF, rounding the final answer to four decimal places.
Method. The empirical conditional CDF at 4 is \(3/5=0.60\). Use \(F_X(4)=F_X(2)+S_X(2)F_X(4\mid X>2)\):
The estimate is 0.6800. Checking the conditional probability gives \((0.6800-0.20)/0.80=0.60\), matching the observed fraction. The common mistake avoided is reporting \(3/5=0.60\) as the original CDF; that fraction estimates the CDF conditional on exceeding 2, not the unconditional probability of a loss at most 4.
Common Traps and Interpretation
- Confusing censoring with truncation. A right-censored record is present and contributes a survival probability. A truncated loss is absent, and its absence changes the distribution conditional on selection.
- Using a density for a threshold statement. A record \(X>c\) contributes \(S_X(c)\), or \(S_X(c)/S_X(a)\) when conditioned on entry at \(a\); it does not contribute \(f_X(c)\).
- Ignoring the entry condition. For an exact loss above a threshold, the conditional density includes \(S_X(a)\) in the denominator. For censoring, the same conditioning is required.
- Using an incorrect risk set. With delayed entry, a claim counts as at risk at time \(t\) only if it has entered by \(t\) and has not already experienced an event or censoring.
- Assuming the unobserved tail is empirically identified. Truncated records do not determine the probability below the cutoff. An external assumption or additional data are needed to recover that part of \(F_X\).
A defensible analysis therefore states the loss model or empirical target, the entry and censoring rules, and the independence assumptions before calculating. These details determine whether a likelihood is conditional, which observations contribute survival probabilities, and who belongs in each empirical risk set.
Check Your Understanding
For each problem, identify the observation mechanism before calculating.
- A loss is observed only if it exceeds 3, and its exact value is 5. Write its conditional likelihood contribution in terms of \(f_X\) and \(S_X\). (Answer: \(f_X(5)/S_X(3)\).)
- Losses are exponential with rate \(\lambda=0.25\). A record is included only if \(X>2\), then is right-censored at 6. Calculate its conditional likelihood contribution to four decimal places. (Answer: \(S_X(6)/S_X(2)=e^{-1}\approx0.3679\).)
- Only losses exceeding \(a=4\) are observed. A reference model gives \(F_X(4)=0.10\), and 4 of 5 observed losses are at most 7. Calculate the adjusted estimate of \(F_X(7)\). (Answer: \(0.10+0.90(4/5)=0.8200\).)
- At an event time, \(R_j=5\) records are at risk and \(d_j=2\) have events. What is the product-limit survival multiplier at that time? (Answer: \(1-2/5=0.6000\).)
- A claim is excluded from a file unless its loss exceeds 1.5. For an included claim, the recorded information is only \(X>3\). State the correct likelihood contribution and explain the role of the denominator. (Answer: \(S_X(3)/S_X(1.5)\); the denominator conditions on exceeding the truncation threshold.)