Assumption Checks for Bühlmann Models

Why Check the Model Before Credibility Weighting?

A Bühlmann estimate combines a class's own experience with a collective mean. That combination is useful only to the extent that the data behave like the model assumes. A numerical credibility factor can be calculated even when class means are drifting, the portfolio mixes unlike populations, or observations are not generated under a consistent variance pattern. The result may be precise arithmetic applied to an unsuitable model.

The purpose of an assumption check is not to prove that a model is true. It is to look for patterns that are difficult to reconcile with the model, decide whether they are material, and consider whether the data should be adjusted, segmented, or modeled differently. These checks support credibility weighting of rates and loss costs and are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: In the Bühlmann model, let $X_{ij}$ be the observation for risk class $i$ in period $j$, and let $\Theta_i$ represent the class's underlying risk characteristic. Conditional on $\Theta_i$, observations have a stable mean and variance: $$ E[X_{ij}\mid\Theta_i]=\mu(\Theta_i), \qquad \operatorname{Var}(X_{ij}\mid\Theta_i)=\sigma^2(\Theta_i). $$ The observations for a class are assumed conditionally independent given $\Theta_i$. The risk characteristics $\Theta_i$ are drawn from a common population. The structural parameters are $\mu=E[\mu(\Theta)]$, the population mean; $\mathrm{EPV}=E[\sigma^2(\Theta)]$, the expected process variance; and $\mathrm{VHM}=\operatorname{Var}(\mu(\Theta))$, the variance of hypothetical means. These quantities must be finite.

The model does not require every class to have the same conditional mean or the same conditional variance. Instead, each class has a stable conditional mean and variance, while the portfolio provides a common distribution of class characteristics from which the structural parameters are defined. For equally weighted observations, the classic Bühlmann formulation is the natural starting point. If observations have materially different exposures, the variance and weighting structure must reflect that difference; the earlier material on Bühlmann-Straub credibility treats unequal weights.

What the Assumptions Imply

Each assumption suggests a different diagnostic. Stability concerns whether a class's expected level changes over time. Common structural parameters concern whether the classes being pooled belong to a sufficiently comparable population. Conditional independence concerns whether one observation supplies information beyond the class characteristic alone. The variance-pattern check asks whether the conditional variation is consistent from period to period for a given class.

A useful starting point is the law of total variance. It explains why the model separates process variation from variation in underlying class means:

\operatorname{Var}(X) =E[\operatorname{Var}(X\mid\Theta)] +\operatorname{Var}(E[X\mid\Theta]) =\mathrm{EPV}+\mathrm{VHM}.

The first term describes variation within a class around its conditional mean; the second describes variation among those conditional means. The identity follows by writing $X-E[X]$ as $(X-E[X\mid\Theta])+(E[X\mid\Theta]-E[X])$. On squaring and taking expectations, the cross-product has expectation zero because $E[X-E[X\mid\Theta]\mid\Theta]=0$. The two remaining terms are exactly the expected conditional variance and the variance of the conditional mean.

This decomposition is informative only if the periods and classes used to estimate the components are reasonably comparable. A trend, a change in exposure mix, or a portfolio segment with a different mean can inflate apparent between-class variation or distort apparent process variation.

A Practical Assumption-Check Sequence

Define the observation and exposure.
Confirm what one $X_{ij}$ represents, its time period, and whether observations have comparable exposure. A change in the unit of observation can create apparent changes in means or variances.

Check stability within classes.
Compare period means or adjusted rates for the same classes. Look for a systematic trend, a step change, or recurring calendar patterns. A changing mean is not the stable conditional mean assumed by the model.

Check comparability across the pooled population.
Compare segments that may have different underlying populations. Differences in their means or variance components can make a single set of structural parameters misleading.

Inspect residual patterns and variation.
After allowing for class means and appropriate exposure, look for repeated patterns across observations and for material differences in spread across periods. Treat these as flags to investigate, not automatic proof that an assumption fails.

Worked Diagnostics

Worked Example: Detecting a Changing Class Mean

A hypothetical insurer reviews annual claim costs, in dollars per exposure, for two classes. Class A records $90, 92, 94, 96$; Class B records $110, 112, 114, 116$. The diagnostic model treats annual observations as conditionally independent normal variables with stable class mean and conditional variance $16$: $X_{ij}\mid\Theta_i\sim N(\mu(\Theta_i),16)$. There is no interest-rate or mortality assumption because this is a claim-cost diagnostic. Determine whether the stable-mean assumption looks plausible.

Step 1: Calculate each class's overall mean. Class A has mean $(90+92+94+96)/4=372/4=93$. Class B has mean $(110+112+114+116)/4=452/4=113$.

Step 2: Compare the first and second halves. For A, the first-half mean is $(90+92)/2=91$, and the second-half mean is $(94+96)/2=95$, a rise of $95-91=4$. For B, the corresponding means are $(110+112)/2=111$ and $(114+116)/2=115$, also a rise of $4$.

Step 3: Check whether the pattern is shared. The portfolio's annual averages are $100,102,104,106$, so they rise by $2$ each year. Both classes move upward by $2$ per year, consistent with a common time effect rather than an isolated class shift.

Conclusion and check. The data show a systematic increase in the observed level, which is a warning against treating the unadjusted class means as stable over all four periods. The stated conditional variance is $16$, or standard deviation $4$, but that does not make the upward pattern disappear; the pattern should be investigated and, where justified, adjusted before estimating credibility. The common mistake is to calculate the four-period means, $93$ and $113$, and assume that one average proves stability. The half-period comparisons show why it does not.

Worked Example: Checking for Common Structural Parameters

Suppose a hypothetical book is divided into two separately administered portfolios. Portfolio A has two fitted class means, $80$ and $100$; Portfolio B has fitted class means, $120$ and $140$. Assume the conditional observations in both portfolios follow $X\mid\Theta\sim N(\Theta,100)$, with independent observations given $\Theta$, and that the within-class variance estimate is $100$ in each. Compare the structural means and VHM in the two portfolios with the values obtained by pooling them equally.

Step 1: Calculate each portfolio's structural mean. For A, $\mu_A=(80+100)/2=90$. For B, $\mu_B=(120+140)/2=130$. The means differ by $40$, a material signal that the portfolios may not share one population mean.

Step 2: Calculate each portfolio's VHM. In A, the deviations from $90$ are $-10$ and $10$, so $\mathrm{VHM}_A=[(-10)^2+10^2]/2=100$. In B, deviations from $130$ are also $-10$ and $10$, so $\mathrm{VHM}_B=100$. Each portfolio's stipulated EPV is $100$.

Step 3: Calculate the equally pooled values. The pooled mean is $(80+100+120+140)/4=110$. The deviations from $110$ are $-30,-10,10,30$. Thus the pooled VHM is $(900+100+100+900)/4=500$. Pooling has raised the apparent VHM from $100$ within either portfolio to $500$, because it includes the gap between portfolio means.

Conclusion and check. A common structural mean is not plausible without a reason to adjust or combine the portfolios. The calculation checks: the pooled mean $110$ lies halfway between $90$ and $130$, and the pooled VHM is larger than either within-portfolio VHM. The common mistake is to treat the pooled VHM of $500$ as pure risk-class variation. Here it also captures the portfolio difference, so a single pooled credibility model can overstate heterogeneity among classes.

Worked Example: Comparing Conditional Variance Across Periods

For a hypothetical class with a stable expected annual cost of $100$, equal exposure is observed in each of two periods. The stipulated diagnostic model is normal: $X\mid\Theta,\text{period 1}\sim N(100,100)$ and $X\mid\Theta,\text{period 2}\sim N(100,400)$. In each period, the observed costs are $90,100,110$ and $80,100,120$, respectively. Observations are assumed independent within each period for this diagnostic. Is a common conditional variance across periods plausible?

Step 1: Confirm the observed means. Period 1 has mean $(90+100+110)/3=100$; Period 2 has mean $(80+100+120)/3=100$. The means agree with the stipulated stable mean.

Step 2: Calculate sample variances. Period 1 deviations from $100$ are $-10,0,10$, with squared deviations summing to $100+0+100=200$. Its sample variance is $200/(3-1)=100$. Period 2 deviations are $-20,0,20$, with squared deviations summing to $400+0+400=800$; its sample variance is $800/(3-1)=400$.

Step 3: Compare the spreads. The period 2 sample variance is $400/100=4$ times the period 1 variance. The assumed normal distributions also have variances $100$ and $400$, respectively, so the specified model itself does not have a common conditional variance across periods.

Conclusion and check. The equal means do not establish a consistent variance pattern. The observed variance ratio is $4$, matching the assumed variance ratio. With only three observations per period, this is a diagnostic signal rather than strong statistical evidence by itself. The common mistake is to check only whether period means agree; credibility estimates also depend on the variation around those means.

Independence, Uncertainty, and Common Traps

Conditional independence means that, after conditioning on the class characteristic $\Theta_i$, one period's observation does not carry additional information about another period's random deviation. It is worth asking whether records overlap, whether the same event affects multiple observations, or whether the data contain recurring patterns not explained by the class characteristic. Such features can make the apparent amount of independent experience larger than the amount actually present. A visible pattern is a prompt for investigation; it is not, on its own, a complete dependence analysis.

All empirical checks are limited by sample size. A small class can have a noisy mean or variance even when the underlying assumptions hold. Conversely, a large data set can make a small but persistent departure visible. Therefore, compare the size and persistence of a pattern with the practical consequences of using the credibility estimate, and avoid treating a single unusual observation as decisive.

Assuming that a pooled mean is stable. A long-run average can hide a trend or a step change. Compare periods within classes.
Pooling unlike populations without checking. A portfolio-level shift can inflate apparent VHM and change the collective mean.
Equating different class variances with an automatic model failure. The model permits $\sigma^2(\Theta)$ to vary by class. The relevant check is whether the variance pattern is consistent with the model and exposure structure being used.
Confusing a warning with a proof. A diagnostic flags a potential mismatch; the analyst must consider sampling variation, data quality, exposure changes, and the business context.

Key takeaway: Before using Bühlmann credibility, check that the class means are stable, the pooled classes plausibly share structural parameters, observations are reasonably consistent with conditional independence, and the variance pattern matches the model. If a check raises concern, identify its source before relying on the resulting credibility weight.

Check Your Understanding

For each hypothetical scenario, identify the assumption at issue and calculate the requested diagnostic.

A class has annual costs $50,52,56,58$, with equal exposure in all years. Compare the mean of the first two years with the mean of the last two. (Answer: $51$ versus $57$, a rise of $6$; stable class mean is questionable.)
Two segments have fitted class means $70,90$ and $110,130$. Find each segment mean and state whether a common structural mean is plausible without adjustment. (Answer: $80$ and $120$; the $40$-unit gap questions a common structural mean.)
For equal-exposure observations in one class, the residuals in period 1 are $-5,0,5$, and those in period 2 are $-10,0,10$. Calculate each sample variance and their ratio. (Answer: $25$ and $100$, with period 2 variance four times period 1.)
Four policies have residuals of $+8$ in a year with a shared hypothetical storm, and four other policies have residuals of $-8$ in a year without one. What assumption should be investigated before treating all eight records as independent information? (Answer: Conditional independence; the common event may create related residuals.)
A portfolio has two equally represented segments with means $60$ and $100$, and within-segment VHM $64$ in each. Calculate the pooled mean and pooled VHM, and identify the interpretation trap. (Answer: Pooled mean $80$; pooled VHM $464$, since $[(60-80)^2+64+(100-80)^2+64]/2=464$; the pooled value includes the segment difference, not only within-segment heterogeneity.)

What You'll Learn