Why Record-Specific Censoring Limits Matter
In a severity study, a loss may be observed exactly if it occurs before the record’s observation limit. If the loss exceeds that limit, the record only tells us that the loss was larger than the limit. Different records can have different observation limits, for example because hypothetical policies have different reporting windows or study periods. Each censored record must therefore contribute information at its own censoring point.
A separate issue arises when a loss enters the data set only if it exceeds an inclusion threshold. That threshold changes the distribution of observed losses: the likelihood must account for the fact that smaller losses could not enter the sample. A censoring limit describes how much is known about an included loss; a truncation threshold governs whether the loss is included at all.
The censoring limits may differ across records. In the likelihood, an exact observation uses its observed loss in the density, while a censored observation uses its own limit in the survival function. The likelihood below conditions on the known limits and assumes that, given the model and any recorded covariates, the censoring mechanism does not convey additional information about the loss. If censoring is informative, these contributions alone do not describe the full likelihood.
Deriving the Censoring Contribution
For an exact observation \(X_i=z_i\), the likelihood contribution is a density, since the observed value is known to lie near \(z_i\). For a censored observation, we do not know the value of \(X_i\); we know only that it exceeded \(c_i\). The probability of that event is the integral of the density over all compatible loss values:
Thus, for independent losses with known censoring limits, the likelihood is
Here \(\theta\) denotes the model parameter or parameter vector. For an exact record, \(z_i=X_i\), and its censoring limit does not replace the observed value in the density. For a censored record, \(z_i=c_i\), and the contribution is the probability of exceeding that limit. If the censoring limit is selected by a process that depends on the unobserved loss, conditioning on the limits may not make the censoring mechanism ignorable.
Worked Example: Exponential Losses With Four Different Limits
Suppose independent hypothetical losses follow an exponential distribution with rate \(\lambda>0\), so \(f_X(x)=\lambda e^{-\lambda x}\) and \(S_X(x)=e^{-\lambda x}\) for \(x\geq 0\). Four records have these observations: exact losses 1 and 2, each observed before its respective limit of 4 and 5; and right-censored losses at limits 2 and 3. Find the maximum likelihood estimate of \(\lambda\).
Method. Use a density for each exact loss and a survival probability at each censored record’s own limit. There are two exact records, so the density terms contribute \(\lambda^2\). Their exponential factors are \(e^{-\lambda}\) and \(e^{-2\lambda}\); the censored records contribute \(e^{-2\lambda}\) and \(e^{-3\lambda}\). Therefore
The log-likelihood is \(\ell(\lambda)=2\log\lambda-8\lambda\). Its derivative is \(2/\lambda-8\), so the score equation gives \(\hat{\lambda}=2/8=0.25\). Since \(\ell''(\lambda)=-2/\lambda^2<0\), this is a maximum. The estimate is \(0.250\) per unit of loss.
Check. Substituting \(\hat{\lambda}=0.25\) into the score gives \(2/0.25-8=0\). The method avoids treating the two censored limits as exact losses.
Adding a Truncation Threshold
Now suppose a record is included only when its loss exceeds a known threshold \(a_i\). This is left truncation: losses at or below \(a_i\) are absent from the observed sample. Conditional on inclusion, the loss distribution has survival denominator \(S_X(a_i)\). For a continuous model, an exact included loss \(x_i>a_i\) therefore contributes \(f_X(x_i)/S_X(a_i)\).
If the included loss is right-censored at \(c_i\), with \(c_i\geq a_i\), the observed information is \(X_i>c_i\). Conditional on the fact that it entered the sample, the probability of this information is \(P(X_i>c_i\mid X_i>a_i)=S_X(c_i)/S_X(a_i)\). Consequently, when both mechanisms operate, the conditional likelihood is
This formula assumes each record is sampled conditional on exceeding its own known inclusion threshold and that \(c_i\geq a_i\). The truncation denominator appears for both exact and censored included records. It is not a replacement for the censoring survival term: the numerator for a censored record still refers to its own censoring limit.
Worked Example: Separate Entry and Censoring Thresholds
Suppose hypothetical losses are independent exponential random variables with rate \(\lambda>0\). A record is included only if \(X_i>a_i\). The data are: an exact loss of 3 with \(a_1=1\) and censoring limit \(c_1=4\); a censored record at \(c_2=4\) with \(a_2=2\); an exact loss of 5 with \(a_3=2\) and \(c_3=6\); and a censored record at \(c_4=3\) with \(a_4=1\). Find the conditional maximum likelihood estimate of \(\lambda\).
Method. For the exponential model, \(f_X(x)/S_X(a)=\lambda e^{-\lambda(x-a)}\) for an exact included loss, and \(S_X(c)/S_X(a)=e^{-\lambda(c-a)}\) for a censored included loss. The four contributions are, in order,
For example, the first exact contribution uses \(3-1=2\), and the second censored contribution uses \(4-2=2\). Multiplying gives \(L(\lambda)=\lambda^2e^{-9\lambda}\), since the total exponential exposure is \(2+2+3+2=9\). Then \(\ell(\lambda)=2\log\lambda-9\lambda\), and the score equation \(2/\lambda-9=0\) yields \(\hat{\lambda}=2/9=0.2222\) per unit of loss, rounded to four decimal places. The second derivative is \(-2/\lambda^2<0\), confirming a maximum.
Check. The exact observations exceed their inclusion thresholds, and each censoring limit is at least its record’s threshold. The calculation uses the entry thresholds in the conditional denominators and the censoring limits in the survival numerators. It avoids treating entry as though it were the point at which observation stopped.
Varying Limits With a Different Severity Model
The record-by-record construction does not depend on choosing an exponential model. The model determines the density and survival functions; the observation status determines which function enters each likelihood contribution. The next example illustrates this distinction with a Weibull model whose shape is fixed.
Worked Example: Fixed-Shape Weibull Model
Suppose hypothetical independent losses have a Weibull distribution with known shape \(k=2\) and unknown scale \(\theta>0\). Thus \(S_X(x)=e^{-(x/\theta)^2}\) and \(f_X(x)=2x\theta^{-2}e^{-(x/\theta)^2}\) for \(x>0\). Two records are exact losses 1 and 2, observed before limits 4 and 5; two others are right-censored at limits 1.5 and 2.5. Find the maximum likelihood estimate of \(\theta\).
Method. Set \(\eta=1/\theta^2\), so the exact density is \(2x\eta e^{-\eta x^2}\) and a censored record at \(c\) contributes \(e^{-\eta c^2}\). The likelihood is
The exponent total is \(1^2+2^2+1.5^2+2.5^2=1+4+2.25+6.25=13.5\). Hence \(\ell(\eta)=\log 8+2\log\eta-13.5\eta\), and \(2/\eta-13.5=0\) gives \(\hat{\eta}=2/13.5=4/27\). Transforming back, \(\hat{\theta}=1/\sqrt{\hat{\eta}}=\sqrt{27/4}\approx2.598\), rounded to three decimals. Since \(\ell''(\eta)=-2/\eta^2<0\), this is a maximum. This construction uses each censored record’s own limit squared; it does not substitute a pooled censoring limit.
Common Errors and Practical Checks
A frequent error is to treat a censored limit as though it were an exact loss. In the first example, that mistake would assign a density to all four recorded values \(1,2,2,3\), giving the incorrect likelihood \(\lambda^4e^{-8\lambda}\) and estimate \(\hat{\lambda}=4/8=0.5\). The correct likelihood has only two density factors, one for each exact loss, and survival factors for the two censored records: \(\lambda^2e^{-8\lambda}\). Its estimate is \(2/8=0.25\). A censored limit identifies an event \(X>c_i\), not an observed loss equal to \(c_i\).
A second error is to use one common censoring limit when limits differ. Each censored record contributes \(S_X(c_i)\) at its own \(c_i\). A third error is to divide by a censoring limit’s survival probability as though it were an inclusion normalizer. Only a truncation rule that determines sample inclusion creates a conditional denominator such as \(S_X(a_i)\). In particular, if a record is included when \(X_i>a_i\) and censored at \(c_i\), its contribution is \(S_X(c_i)/S_X(a_i)\), not \(S_X(a_i)/S_X(c_i)\) or \(S_X(c_i)\) alone.
For each record, decide whether the loss is exact or right-censored; do not infer status from the size of a recorded number alone.
For an exact record, write the density at the observed loss. For a censored record, write the survival probability at that record’s censoring limit.
If the sample includes only losses above a threshold, divide each applicable contribution by the probability of exceeding that threshold.
Confirm that every exact observation exceeds its entry threshold, every censoring limit is feasible under the design, and the number of density factors matches the number of exact observations.
These checks matter because censoring and truncation discard different information. Censoring replaces an exact value with a bound for a loss that is in the sample. Truncation removes losses from the sample according to an inclusion rule. Keeping those mechanisms distinct makes the likelihood interpretable and is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.
Check Your Understanding
For each problem, write the relevant likelihood contribution and check whether the final estimate uses the correct censoring or truncation information.
- Independent losses follow an exponential model with rate \(\lambda\). One loss is observed exactly as 2 before its limit of 5, and two others are censored at 1 and 4. Find the maximum likelihood estimate of \(\lambda\). (Answer: \(1/7\approx0.1429\))
- Exponential losses with rate \(\lambda\) are included only above their stated entry thresholds. The records are: exact 4 with entry 1 and limit 6; censored at 5 with entry 2; exact 3 with entry 1 and limit 4. Find the conditional maximum likelihood estimate. (Answer: \(0.250\))
- A Weibull model has known shape \(k=2\) and unknown scale \(\theta\). Exact losses are 2 and 3; other records are censored at 1 and 4. Find the maximum likelihood estimate of \(\theta\). (Answer: \(\sqrt{15}\approx3.873\))
- Exponential losses are included only when they exceed their entry thresholds. There is an exact loss 5 with entry 2, a censored record at 6 with entry 1, an exact loss 4 with entry 1, and a censored record at 3 with entry 2. Find the conditional maximum likelihood estimate of the rate. (Answer: \(1/6\approx0.1667\))