Why the Observed Data Determine the Likelihood
A loss record does not always identify the loss amount exactly. A policy may still be open at the valuation date, so the recorded information is only that the loss exceeds the amount paid or observed so far. A reporting threshold may also mean that a loss enters the data set only after exceeding a specified amount. These are different observation mechanisms, and each changes the likelihood used to fit a severity model.
AIC and BIC are calculated from a model's maximized log-likelihood, but that log-likelihood must describe the observations actually available. Treating a bound or censoring time as if it were an exact loss generally gives a different, inappropriate likelihood. The resulting information criterion then assesses the wrong data model.
The usual likelihood contributions below assume that censoring is non-informative: conditional on the model and any stated covariates, the censoring mechanism does not provide additional information about the unobserved loss. For left truncation, the entry thresholds are treated as known, and the analysis conditions on the losses having passed those thresholds. If these assumptions are not appropriate, the observation or selection mechanism may need to be modelled explicitly.
Likelihood Contributions for Incomplete Observations
Let \(X\) be a nonnegative loss with density \(f_X\), distribution function \(F_X\), and survival function \(S_X(x)=1-F_X(x)\). Let \(\theta\) denote the model parameters. For independent observations, the likelihood is the product of the applicable contribution for each record; the log-likelihood is the sum of the log contributions.
| Recorded information | Likelihood contribution | Reason |
|---|---|---|
| Exact loss \(x\) | \(f_X(x;\theta)\) | The observed value contributes its density. |
| Loss exceeds \(c\) | \(S_X(c;\theta)\) | The event known to have occurred is \(X>c\). |
| \(a<X\le b\) | \(F_X(b;\theta)-F_X(a;\theta)\) | The observed event is that the loss falls in the interval. |
| Exact \(x\) observed only when \(X>a\) | \(f_X(x;\theta)/S_X(a;\theta)\), for \(x>a\) | The likelihood is conditional on being included. |
The interval contribution can also be written \(P(a<X\le b)\). It is a probability, not a density evaluated at an interval endpoint. For an exact loss \(x\) subject to a known left-truncation threshold \(a\), the conditional density follows from the definition of conditional probability:
The denominator is necessary because the observed sample excludes losses that did not exceed \(a\). Without it, the likelihood describes an unselected sample rather than the distribution of losses conditional on inclusion.
For independent records, the complete observed-data likelihood is the product of these contributions. If a record is both subject to a known entry threshold and censored after entry, its contribution must be conditioned on inclusion as well. For example, observing only that \(X>c\) among losses known to exceed \(a\), where \(c\ge a\), contributes \(S_X(c;\theta)/S_X(a;\theta)\).
Information Criteria After Fitting
For a candidate model with \(k\) free parameters, let \(\ell(\widehat{\theta})\) be the maximized log-likelihood formed from the appropriate observed-data contributions. The standard criteria are:
Here, \(n\) is the number of independent observed units represented in the likelihood. In a one-record-per-loss analysis, it is the number of included loss records, not the number of exact losses alone. A censored record remains an observed unit and contributes information through its censoring event. For truncated data, \(n\) is the number of included observations under the stated conditional likelihood. For both criteria, smaller values are preferred among candidates fitted to the same observed data under comparable observation assumptions.
The formulas do not repair an incorrectly constructed likelihood. AIC or BIC calculated after treating censored values as exact may be arithmetically correct but statistically answer a different question. Candidate likelihoods must also include consistent constants and use the same data and observation scheme for a meaningful comparison.
Worked Applications
Worked Example: Right-Censored Exponential Losses
A hypothetical coverage portfolio has three independent loss records under a constant-force exponential severity model, \(X\sim\operatorname{Exponential}(\lambda)\), with density \(f_X(x)=\lambda e^{-\lambda x}\) and survival function \(S_X(x)=e^{-\lambda x}\), for \(x\ge0\). Two exact losses are \(2\) and \(5\) thousand dollars. The third record is right-censored at \(3\) thousand dollars: it is known only that the loss exceeds \(3\). Fit \(\lambda\), then calculate AIC and BIC.
Method. Use densities for the two exact losses and a survival probability for the censored record. The likelihood is \(L(\lambda)=f_X(2)f_X(5)S_X(3)\). Thus \(L(\lambda)=\lambda e^{-2\lambda}\lambda e^{-5\lambda}e^{-3\lambda}=\lambda^2e^{-10\lambda}\).
The log-likelihood is \(\ell(\lambda)=2\ln\lambda-10\lambda\). Its derivative is \(2/\lambda-10\), so setting it equal to zero gives \(\widehat{\lambda}=2/10=0.2\) per thousand dollars. The second derivative is \(-2/\lambda^2<0\), confirming a maximum.
Substitution gives \(\ell(\widehat{\lambda})=2\ln(0.2)-10(0.2)=2(-1.609438)-2=-5.218876\). Therefore, \(-2\ell=10.437752\). With \(k=1\) and \(n=3\), AIC is \(10.437752+2(1)=12.437752\), and BIC is \(10.437752+1\ln(3)=10.437752+1.098612=11.536364\). To four decimal places, AIC is \(12.4378\) and BIC is \(11.5364\).
Check. The censored record contributes \(P(X>3)=e^{-3\lambda}\); the fitted likelihood contains three record-level contributions, so \(n=3\). The common error is to use \(f_X(3)\) for the censored record, which falsely asserts that its loss equals \(3\).
Worked Example: Exact Losses From a Left-Truncated Sample
A hypothetical data set includes only losses that exceed their known entry thresholds. The severity model is exponential with density \(f_X(x)=\lambda e^{-\lambda x}\) and survival \(S_X(a)=e^{-\lambda a}\). One included exact loss is \(3\) thousand dollars with threshold \(a=1\); another is \(5\) thousand dollars with threshold \(a=2\). Calculate the conditional maximum-likelihood estimate, AIC, and BIC.
Method. For each exact included loss \(x\), use \(f_X(x)/S_X(a)\), not the unconditioned density. For an exponential model, \(f_X(x)/S_X(a)=\lambda e^{-\lambda(x-a)}\). The likelihood is therefore \(\lambda e^{-\lambda(3-1)}\lambda e^{-\lambda(5-2)} =\lambda^2e^{-5\lambda}\).
The log-likelihood is \(2\ln\lambda-5\lambda\), with derivative \(2/\lambda-5\). The maximum occurs at \(\widehat{\lambda}=2/5=0.4\) per thousand dollars. The second derivative, \(-2/\lambda^2\), is negative. The maximized log-likelihood is \(2\ln(0.4)-5(0.4)=2(-0.916291)-2=-3.832582\). Thus \(-2\ell=7.665164\).
There are two included independent records, so \(n=2\); there is one estimated parameter, so \(k=1\). AIC is \(7.665164+2=9.665164\). BIC is \(7.665164+\ln(2)=7.665164+0.693147=8.358311\). To four decimal places, AIC is \(9.6652\) and BIC is \(8.3583\).
Check. The exponential conditional density depends on the excess above the entry threshold, giving total excess \( (3-1)+(5-2)=5\). Ignoring truncation would instead use \(\lambda^2e^{-8\lambda}\), based on the raw total loss \(3+5=8\). That likelihood describes unselected exact losses and does not match how these records entered the sample.
Worked Example: Comparing Fitted Models With Mixed Censoring
A hypothetical portfolio has six independent included records: three exact losses, one interval-censored loss known to lie in \((4,7]\) thousand dollars, and two right-censored losses known to exceed \(8\) and \(10\) thousand dollars. Assume non-informative censoring and no truncation. Maximum-likelihood fitting using the correct contributions gives maximized log-likelihood \(-8.40\) for an exponential model with \(k=1\), and \(-7.95\) for a Weibull model with \(k=2\). Calculate and compare AIC and BIC.
Method. The interval-censored record contributes \(F_X(7)-F_X(4)\); the two right-censored records contribute \(S_X(8)\) and \(S_X(10)\); each exact record contributes its density. Each model's stated maximized log-likelihood is for the product of these six contributions. Use \(n=6\) and each model's parameter count.
For the exponential model, \(-2\ell=-2(-8.40)=16.80\). Therefore, AIC is \(16.80+2(1)=18.80\). Since \(\ln(6)=1.791759\), BIC is \(16.80+1(1.791759)=18.591759\), or \(18.5918\) to four decimal places.
For the Weibull model, \(-2\ell=-2(-7.95)=15.90\). Therefore, AIC is \(15.90+2(2)=19.90\). BIC is \(15.90+2(1.791759)=15.90+3.583518=19.483518\), or \(19.4835\) to four decimal places.
Both criteria are lower for the exponential fit: its AIC is \(18.80\) rather than \(19.90\), and its BIC is \(18.5918\) rather than \(19.4835\). This comparison is meaningful because the models use the same six records and the same censoring information. The likelihood figures are fitted outputs supplied for this calculation; they must be obtained by maximizing the likelihood with the interval probability and survival contributions, not by replacing interval endpoints or censoring thresholds with exact losses.
Common Traps and Practical Checks
A useful check is to ask, for every record, “What event does the data actually establish?” An exact value establishes a density contribution; censoring establishes an event or interval; truncation changes the population being conditioned on. The information criterion should be the final step after that question is answered for every observation.
- Using a density at the censoring threshold. A record known to exceed \(c\) contributes \(S_X(c)\), not \(f_X(c)\). The threshold is not an observed exact loss.
- Using an interval endpoint as the observation. If \(a<X\le b\), contribute \(F_X(b)-F_X(a)\). Neither endpoint is the measured loss.
- Forgetting the truncation denominator. Exact included values contribute \(f_X(x)/S_X(a)\) when inclusion requires \(X>a\). Omitting \(S_X(a)\) ignores selection.
- Dropping censored records from \(n\). A censored record still contributes information to the likelihood and counts as an observed unit in the stated independent-record setting.
- Comparing different observed data. If candidate likelihoods use different records or different handling of censoring, the criterion values do not provide the intended like-for-like comparison.
Correctly handling censoring and truncation is essential to defensible severity-model selection and is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.
Check Your Understanding
For each problem, identify the observation contribution before applying any criterion.
- Under an exponential model with rate \(\lambda\), one loss is known only to exceed \(c=4\) thousand dollars. Write its likelihood contribution. (Answer: \(S_X(4)=e^{-4\lambda}\).)
- An exact loss \(x=6\) thousand dollars is included only if it exceeds a known threshold \(a=2\). Under an exponential model, write its conditional density contribution. (Answer: \(f_X(6)/S_X(2)=\lambda e^{-4\lambda}\).)
- A loss is known to lie in \((3,8]\). Give its likelihood contribution in terms of \(F_X\). (Answer: \(F_X(8)-F_X(3)\).)
- A fitted model has \(\ell(\widehat{\theta})=-12\), \(k=2\), and \(n=5\) independent included records. Calculate AIC and BIC to four decimal places. (Answer: AIC \(=28.0000\); BIC \(=27.2189\).)
- Two models use the same seven records, including censored records. Their maximized log-likelihoods are \(-10.2\) with \(k=1\) and \(-9.5\) with \(k=3\). Calculate both AIC values and state which is lower. (Answer: \(22.4\) and \(25.0\), respectively; the first model has the lower AIC.)