Worked AIC And BIC Comparison Of Severity Families

Comparing Fitted Severity Families

A fitted severity model can summarize the same loss data in different ways. An exponential model is parsimonious, while gamma and lognormal models can represent a broader range of positive-loss shapes. Comparing their maximized log likelihoods alone favors the model that fits best in-sample, but that model may also use more parameters. The Akaike information criterion (AIC) and Bayesian information criterion (BIC) balance fit against a complexity penalty.

This tutorial applies both criteria to fitted exponential, gamma, and lognormal severity candidates. The calculations assume that each candidate was fitted to the same $n$ observed losses using the same likelihood convention. This comparability matters: differences in observations, truncation treatment, censoring treatment, or likelihood constants can make a numerical ranking misleading.

Definition: Let $\hat{\ell}$ be a model's maximized log likelihood, $k$ its number of freely estimated parameters, and $n$ the number of independent observations used in the likelihood. Then $$ \mathrm{AIC}=-2\hat{\ell}+2k,\qquad \mathrm{BIC}=-2\hat{\ell}+k\ln(n). $$ For either criterion, the smaller value is preferred among the candidates being compared. AIC and BIC are relative selection criteria; their absolute values are not measures of a model's probability of being true.

Parameter Counts and Calculation Setup

For the usual two-parameter gamma model, both the shape and scale are estimated, so $k=2$. A usual lognormal model estimates its log-scale mean and standard deviation, also giving $k=2$. An exponential model with its rate estimated has $k=1$. These counts presume that no parameter is fixed or estimated separately from the data. If a model has additional freely estimated parameters, they must be included in $k$.

The maximized log likelihood is normally negative for continuous severity data, but its sign does not change the rule: use $-2\hat{\ell}$, then add the relevant penalty. Because $\hat{\ell}$ is maximized, a less negative value indicates a higher maximized likelihood. AIC adds $2k$; BIC adds $k\ln(n)$. For $n>e^2$, BIC's per-parameter penalty exceeds AIC's, although the actual rankings still depend on the fit differences as well as the parameter counts.

The following table summarizes the standard counts used in the examples. The counts apply to the stated model forms, not every possible variant of these families.

Severity candidate	Typical freely estimated parameters	$k$
Exponential with estimated rate	Rate	1
Gamma with estimated shape and scale	Shape, scale	2
Lognormal with estimated log-scale mean and standard deviation	Mean, standard deviation	2

Why the Criteria Use Different Penalties

Both criteria start from the same fit term, $-2\hat{\ell}$. AIC's $2k$ penalty is a large-sample correction for the optimism that arises when a model is evaluated on the data used to estimate it. Under regularity conditions, the expected in-sample improvement from fitting $k$ parameters is approximately offset by this penalty.

BIC's penalty has a different justification. For independent observations in a regular, identifiable model with an interior parameter estimate, a smooth likelihood, and a prior density that is positive near the maximum, a Laplace approximation to the integrated likelihood gives the following leading terms:

\log \{\text{integrated likelihood}\} \approx \hat{\ell}-\frac{k}{2}\ln(n)+O(1).

Multiplying by $-2$ gives $-2\hat{\ell}+k\ln(n)+O(1)$. BIC keeps the leading fit and sample-size penalty terms and omits the bounded remainder. This explains why its penalty grows with sample size. These justifications rely on regularity conditions; the formulas should not be treated as universal exact probabilities or as substitutes for checking whether the models and data are comparable.

Worked Comparisons

Worked Example: Ranking Three Families on the Same Loss Sample

A hypothetical insurer fits three severity distributions to the same $n=100$ positive losses. The fitted exponential model has $k=1$ and $\hat{\ell}=-220.0$; the gamma model has $k=2$ and $\hat{\ell}=-215.5$; the lognormal model has $k=2$ and $\hat{\ell}=-216.8$. Calculate both criteria and identify the preferred candidate under each.

Method and setup. The models use the same 100 observations and comparable maximized log likelihoods. Use $\ln(100)=4.6052$, rounded to four decimal places for display. The unrounded log value is used in the calculations.

Exponential. Its fit term is $-2(-220.0)=440.0$. Thus $\mathrm{AIC}=440.0+2(1)=442.0$, and $\mathrm{BIC}=440.0+1\ln(100)=444.6052$.

Gamma. Its fit term is $-2(-215.5)=431.0$. Thus $\mathrm{AIC}=431.0+2(2)=435.0$, and $\mathrm{BIC}=431.0+2\ln(100)=431.0+9.2103=440.2103$.

Lognormal. Its fit term is $-2(-216.8)=433.6$. Thus $\mathrm{AIC}=433.6+2(2)=437.6$, and $\mathrm{BIC}=433.6+2\ln(100)=433.6+9.2103=442.8103$.

The gamma model has the lowest AIC, 435.0, and the lowest BIC, 440.2103, so both criteria prefer it here. As a check, gamma's maximized log likelihood is the largest (least negative), and its two-parameter penalty still leaves its criterion values below those of the alternatives. The common mistake this calculation avoids is selecting solely by the largest log likelihood without applying the parameter penalty.

Worked Example: AIC and BIC Give Different Rankings

A hypothetical study fits the same three severity families to $n=25$ losses. The exponential model has $k=1$, $\hat{\ell}=-63.0$; the gamma model has $k=2$, $\hat{\ell}=-60.9$; and the lognormal model has $k=2$, $\hat{\ell}=-60.6$. Calculate both criteria and state what the difference in rankings means.

Method and setup. Here $\ln(25)=3.2189$, to four decimal places. Each fit term is calculated as $-2\hat{\ell}$, not $2\hat{\ell}$.

Exponential. The fit term is $126.0$. Therefore, $\mathrm{AIC}=126.0+2=128.0$, and $\mathrm{BIC}=126.0+3.2189=129.2189$.

Gamma. The fit term is $121.8$. Therefore, $\mathrm{AIC}=121.8+4=125.8$, and $\mathrm{BIC}=121.8+2(3.2189)=128.2378$.

Lognormal. The fit term is $121.2$. Therefore, $\mathrm{AIC}=121.2+4=125.2$, and $\mathrm{BIC}=121.2+2(3.2189)=127.6378$.

The lognormal model is preferred by both criteria: its AIC is $125.2$, and its BIC is $127.6378$, each the smallest in its column. Its fit improvement over gamma is modest: the fit term decreases by $0.6$, while the two models have the same $k$, so both criteria preserve that ordering. The exponential model has fewer parameters but a worse fit term. This example illustrates the calculation rather than a universal preference for one severity family. The check is that the lognormal has both the highest maximized log likelihood and the same parameter count as gamma.

Worked Example: How the Sample Size Changes the BIC Penalty

For a separate hypothetical portfolio, all three candidates are fitted to the same $n=400$ losses. The exponential model has $k=1$, $\hat{\ell}=-812.0$; the gamma model has $k=2$, $\hat{\ell}=-808.0$; and the lognormal model has $k=2$, $\hat{\ell}=-809.5$. Calculate AIC and BIC. Explain the role of the larger sample size.

Method and setup. Use $\ln(400)=5.9915$, to four decimal places. The exponential, gamma, and lognormal fit terms are, respectively, $-2(-812.0)=1624.0$, $-2(-808.0)=1616.0$, and $-2(-809.5)=1619.0$.

Exponential. $\mathrm{AIC}=1624.0+2(1)=1626.0$. Also, $\mathrm{BIC}=1624.0+1(5.9915)=1629.9915$.

Gamma. $\mathrm{AIC}=1616.0+2(2)=1620.0$. Also, $\mathrm{BIC}=1616.0+2(5.9915)=1616.0+11.9830=1627.9830$.

Lognormal. $\mathrm{AIC}=1619.0+2(2)=1623.0$. Also, $\mathrm{BIC}=1619.0+11.9830=1630.9830$.

The gamma model has the smallest AIC and BIC. At this sample size, BIC adds $5.9915$ per parameter, compared with AIC's $2$ per parameter. Thus, the two-parameter models each receive a BIC penalty of $11.9830$, while the exponential receives $5.9915$. The larger BIC penalty does not change the winner in these data, but it can change a ranking when a more complex model's fit gain is small. The check is that gamma's fit term is the smallest and its criteria remain smallest after the penalties are added.

Reading the Results Without Overclaiming

A useful comparison reports the criterion values and the ranking, then describes the result conditionally: “Among these candidates, fitted to these observations, gamma has the smallest AIC and BIC.” It does not establish that gamma is the true loss-generating distribution. The criteria compare candidates under their specified likelihoods and parameter counts; they do not repair a poor model set or replace actuarial judgment about tail behavior and intended use.

AIC and BIC can disagree because their penalties differ. For a candidate with $k$ parameters, the AIC penalty is $2k$, while the BIC penalty is $k\ln(n)$. When comparing a one-parameter and a two-parameter model, the extra parameter costs $2$ AIC units but $\ln(n)$ BIC units. A better fit, expressed as a reduction in $-2\hat{\ell}$, must be considered against that extra cost. The calculations above use the standard counts for the stated forms; a fixed parameter would change the count and therefore the criteria.

Before accepting a ranking, confirm that every candidate uses the same observations and the same likelihood scale, that log likelihoods are maximized rather than evaluated at arbitrary estimates, and that $k$ counts all and only the free fitted parameters. In particular, density likelihoods depend on the units in which losses are measured. A consistent change of units changes log likelihoods by the same additive amount per observation for all candidates fitted to those data; mixing likelihood conventions can instead distort a comparison.

Common Calculation Traps

Using the wrong sign. If $\hat{\ell}=-215.5$, the fit term is $-2\hat{\ell}=431.0$, not $-431.0$.
Using the wrong penalty. AIC uses $2k$; BIC uses $k\ln(n)$. Do not use $2k$ for both.
Counting parameters by family name alone. Count parameters that are freely estimated in the particular fitted model. Fixed parameters are not free estimates.
Comparing incomparable likelihoods. A ranking is not meaningful if candidates use different observations or inconsistent treatment of censoring, truncation, or likelihood constants.
Choosing the largest criterion. Among the candidates, a smaller AIC or BIC is preferred, because the criterion combines fit and penalty.

Key takeaway: For each fitted severity candidate, calculate $-2\hat{\ell}$, add $2k$ for AIC or $k\ln(n)$ for BIC, and choose the smallest value within the comparison. State the data and parameter-count assumptions behind that ranking.

Check Your Understanding

For each problem, calculate the requested criteria using the stated parameter counts. Unless otherwise noted, all candidates are fitted to the same observations with comparable maximized log likelihoods.

A hypothetical sample has $n=50$. The exponential model has $k=1$, $\hat{\ell}=-100.0$; gamma has $k=2$, $\hat{\ell}=-97.8$; and lognormal has $k=2$, $\hat{\ell}=-98.0$. Calculate AIC and BIC for all three and name the preferred model under each. (Answer: AIC values are 202.0, 199.6, and 200.0; BIC values are 203.9120, 203.4240, and 203.8240, respectively; gamma is preferred by both.)
For $n=16$, exponential, gamma, and lognormal maximized log likelihoods are $-41.2$, $-39.6$, and $-39.4$, with $k=1,2,2$, respectively. Which model has the smallest AIC and which has the smallest BIC? (Answer: Lognormal for both; its AIC is 82.8 and BIC is approximately 84.3452.)
A fitted exponential model has $n=100$, $k=1$, and $\hat{\ell}=-240.0$. A gamma model has $k=2$ and $\hat{\ell}=-237.2$. Calculate AIC and BIC for each and identify the winner under each criterion. (Answer: Exponential AIC 482.0 and BIC 484.6052; gamma AIC 478.4 and BIC 483.6103; gamma wins both.)
For $n=25$, fitted exponential, gamma, and lognormal models have $k=1,2,2$ and maximized log likelihoods $-63.0$, $-60.9$, and $-60.6$, respectively. Does the best fit by maximized log likelihood also have the smallest BIC? (Answer: Yes. Lognormal has the largest maximized log likelihood and the smallest BIC, approximately 127.6378.)
A study has $n=64$. The exponential, gamma, and lognormal models have $k=1,2,2$, and maximized log likelihoods $-145.0$, $-141.8$, and $-141.6$, respectively. Calculate both criteria and give the selected model under each. (Answer: AIC values are 292.0, 287.6, and 287.2; BIC values are approximately 294.1589, 291.9178, and 291.5178; lognormal is preferred by both.)

What You'll Learn