Common Errors in Structural Parameter Estimation

Why Structural Estimates Are Easy to Misread

In Bühlmann and Bühlmann-Straub credibility, structural parameter estimates are built from two distinct sources of variation. Observations for the same risk vary because of process variation; different risks have different underlying expected outcomes, producing between-risk variation. Confusing these sources, or weighting them incorrectly, can materially distort the credibility parameters.

This tutorial focuses on checking those estimates. It uses the standard Bühlmann-Straub model and its balanced special case. There is no interest or mortality assumption in these examples: they concern risk-level observations and exposures. The methods are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Definition: Let $\Theta_i$ denote the risk parameter for risk $i$, and let $X_{ij}$ be its observed rate in period $j$, with known positive exposure weight $w_{ij}$. Under the Bühlmann-Straub model, conditional on $\Theta_i$, observations are independent and $$ E[X_{ij}\mid\Theta_i]=m(\Theta_i),\qquad \operatorname{Var}(X_{ij}\mid\Theta_i)=\frac{v(\Theta_i)}{w_{ij}}. $$ The risks are independent and share the same structural distribution. The expected process variance (EPV) is $\phi=E[v(\Theta)]$; the variance of hypothetical means (VHM) is $a=\operatorname{Var}(m(\Theta))$. Thus $\phi$ describes variation within a risk, while $a$ describes variation among risk-specific means.

The law of total variance makes the distinction precise. For an observation with exposure weight $w$,

\operatorname{Var}(X)=E[\operatorname{Var}(X\mid\Theta)]+\operatorname{Var}(E[X\mid\Theta]) =\frac{\phi}{w}+a.

A sample spread across risks therefore contains both components. It is not, by itself, an estimate of either component unless the other is accounted for.

Structural Estimates and Their Roles

For risk $i$, let $n_i$ be its number of observations and define its total exposure $w_i=\sum_{j=1}^{n_i}w_{ij}$. The exposure-weighted risk mean and portfolio mean are

\bar X_i=\frac{\sum_j w_{ij}X_{ij}}{w_i},\qquad W=\sum_i w_i,\qquad \bar X=\frac{\sum_i w_i\bar X_i}{W}.

With $r$ risks and $N=\sum_i n_i$ observations, standard method-of-moments estimators are

\widehat{\operatorname{EPV}} =\frac{\sum_i\sum_j w_{ij}(X_{ij}-\bar X_i)^2}{N-r},

\widehat{\operatorname{VHM}} =\frac{\displaystyle \sum_i w_i(\bar X_i-\bar X)^2-(r-1)\widehat{\operatorname{EPV}}} {\displaystyle W-\frac{\sum_i w_i^2}{W}}.

The EPV estimator uses deviations within risks. The VHM estimator starts with weighted variation among risk means, then removes the part expected from process noise. The denominators are not interchangeable: $N-r$ is the within-risk degrees of freedom, while $W-\sum_i w_i^2/W$ is the exposure-weighted between-risk factor.

Why the VHM Process Correction Is Necessary

Write the numerator’s between-risk quantity as $T=\sum_i w_i(\bar X_i-\bar X)^2$. Under the model, $\bar X_i$ has mean $E[m(\Theta)]$ and variance $a+\phi/w_i$. The weighted centering in $T$ removes one fitted overall mean. Taking expectations gives

E[T] =\sum_i\left(w_i-\frac{w_i^2}{W}\right) \left(a+\frac{\phi}{w_i}\right) =a\left(W-\frac{\sum_i w_i^2}{W}\right)+(r-1)\phi.

The $a$-component is multiplied by the VHM denominator. The process component contributes $(r-1)\phi$, because $\sum_i(1-w_i/W)=r-1$. Subtracting $(r-1)\widehat{\operatorname{EPV}}$ therefore corrects the between-risk spread for sampling noise in the risk means.

For the within-risk estimator, the weighted residual sum for risk $i$ has expected value $(n_i-1)\phi$: estimating that risk’s mean uses one degree of freedom. Summing over risks gives $(N-r)\phi$, which explains the EPV denominator.

Worked Calculations and Error Checks

Worked Example: Balanced Data and the Missing Correction

A hypothetical portfolio has three risks, each observed in two unit-exposure periods. The observed rates are $2,4$; $5,7$; and $8,10$. Assume independent conditional observations with $X_{ij}\mid\Theta_i\sim N(\Theta_i,\phi)$, and $\Theta_i\sim N(\mu,a)$; $\mu,\phi,a$ are unknown. Estimate EPV and VHM by the method of moments. No interest or mortality assumptions apply.

Step 1: Estimate EPV. The risk means are $3,6,9$. The within-risk sum of squares is $ (2-3)^2+(4-3)^2+(5-6)^2+(7-6)^2+(8-9)^2+(10-9)^2=6$. Here $N=6$ and $r=3$, so

\widehat{\operatorname{EPV}}=\frac{6}{6-3}=2.

Step 2: Calculate the weighted between-risk spread. Each risk has total exposure $2$; the portfolio mean is $(3+6+9)/3=6$, since the exposures are equal. Therefore $T=2(3-6)^2+2(6-6)^2+2(9-6)^2=36$.

Step 3: Correct and divide. Here $W=6$, $\sum_i w_i^2=12$, and the denominator is $6-12/6=4$. The process correction is $(3-1)(2)=4$. Thus

\widehat{\operatorname{VHM}}=\frac{36-4}{4}=8.

Check. Substituting the estimates into the expected between-risk spread gives $8(4)+2(2)=36$, matching $T$. A common error is to omit the process correction and report $36/4=9$. That assigns within-risk noise in the risk means to VHM; the corrected estimate is $8$.

Worked Example: Unequal Exposures and the Weighted Mean

A hypothetical portfolio has three risks, each with two observations. For each risk the two period rates are one unit below and one unit above its stated mean: $4\pm1$, $6\pm1$, and $8\pm1$. The exposure weights for the two observations of each risk are respectively $0.5,0.5$; $1,1$; and $1.5,1.5$. Assume independent conditional observations with $X_{ij}\mid\Theta_i\sim N(\Theta_i,\phi/w_{ij})$, and $\Theta_i\sim N(\mu,a)$; the structural parameters are unknown. Estimate EPV and VHM. No interest or mortality assumptions apply.

Step 1: Estimate EPV. The weighted within-risk sum of squares for a risk is its total exposure multiplied by $1^2$, because both deviations have magnitude $1$. The three contributions are $1,2,3$, so the total is $6$. Since $N=6$ and $r=3$,

\widehat{\operatorname{EPV}}=\frac{6}{3}=2.

Step 2: Weight the risk means. The total exposures are $w_1=1,w_2=2,w_3=3$, so $W=6$ and $\bar X=(1(4)+2(6)+3(8))/6=40/6=20/3$. The weighted between-risk sum is

T=1(4-20/3)^2+2(6-20/3)^2+3(8-20/3)^2 =\frac{64}{9}+\frac{8}{9}+\frac{48}{9} =\frac{120}{9}=\frac{40}{3}.

Step 3: Apply the correction and denominator. The denominator is $W-\sum_iw_i^2/W=6-(1+4+9)/6=11/3$; the process correction is $(3-1)(2)=4$. Hence

\widehat{\operatorname{VHM}} =\frac{40/3-4}{11/3} =\frac{28/3}{11/3} =\frac{28}{11} \approx 2.5455.

Check. The corrected between-risk sum is $(28/11)(11/3)+4=28/3+12/3=40/3$, as observed. An unweighted mean of the risk means would be $6$, not $20/3$, and would give the wrong between-risk spread. The displayed weighted squared deviations sum to $40/3$, not $80/3$; checking each term prevents a consequential arithmetic error.

Worked Example: Distinguishing the Two Variance Components

Three hypothetical risks each have two unit-exposure observations. Their pairs are $8,12$; $12,16$; and $16,20$, with risk means $10,14,18$. Assume independent conditional observations with $X_{ij}\mid\Theta_i\sim N(\Theta_i,\phi)$, and $\Theta_i\sim N(\mu,a)$; $\mu,\phi,a$ are unknown. Estimate EPV and VHM, then compare them with the marginal variance decomposition. No interest or mortality assumptions apply.

Within-risk calculation. In each pair the deviations from its mean are $-2,2$, giving squared deviations $4+4=8$. The total is $24$, and $N-r=6-3=3$. Thus $\widehat{\operatorname{EPV}}=24/3=8$.

Between-risk calculation. The overall mean is $(10+14+18)/3=14$. Each risk has total exposure $2$, so $T=2(10-14)^2+2(14-14)^2+2(18-14)^2=64$. The VHM denominator is $6-12/6=4$, and the correction is $(3-1)(8)=16$. Therefore

\widehat{\operatorname{VHM}}=\frac{64-16}{4}=12.

Check and interpretation. The estimated marginal variance for a unit-exposure observation is $8+12=20$, by the law of total variance. The raw sample variance of these six observations is $88/5=17.6$; it need not equal the estimated structural marginal variance in this small sample. EPV is not VHM: $8$ measures process variation, while $12$ measures risk-mean variation. Reporting $64/4=16$ as VHM would omit the process correction and overstate it.

Common Errors to Catch Before Reporting Estimates

Mixing within-risk and between-risk variation. Use deviations from $\bar X_i$ for EPV; use deviations of $\bar X_i$ from $\bar X$ for the VHM calculation.
Omitting the process correction. Subtract $(r-1)\widehat{\operatorname{EPV}}$ from the weighted between-risk sum before dividing by its denominator.
Ignoring exposure weights. Use exposure-weighted risk means, the exposure-weighted portfolio mean, and $W-\sum_iw_i^2/W$. An ordinary mean is appropriate only when the relevant exposures are equal.
Confusing EPV with VHM. EPV is an average conditional variance; VHM is a variance of conditional means. A total or pooled variance combines components rather than identifying either one on its own.
Using the wrong degrees of freedom. The EPV denominator is $N-r$, while the correction in the VHM numerator uses $r-1$. Neither count is the VHM denominator.
Rounding too early. Retain exact fractions through the weighted mean, squared deviations, correction, and division. Round only the final estimate.

A reliable audit is to calculate each risk mean and within-risk sum first, then recompute the weighted portfolio mean and between-risk sum independently. Finally, check that the uncorrected between-risk quantity equals the estimated VHM contribution plus the process-noise correction.

Check Your Understanding

For each problem, identify which variation is being estimated and apply the required correction and weights.

Four risks have eight observations in total, and their combined within-risk sum of squares is $18$, with unit weights. What is the EPV estimate? (Answer: $4.5$)
A model has estimated EPV $3.2$ and VHM $5.8$. For a unit-exposure observation, what marginal variance follows from the law of total variance? (Answer: $9.0$)
Three risk means are $3,5,7$, with total exposures $1,2,1$, respectively. The EPV estimate is $1$. Calculate the VHM estimate. (Answer: $2.4$)
Three balanced risks each have total exposure $2$. Their means are $2,5,8$, and the EPV estimate is $1$. The weighted between-risk sum is $36$. Calculate the corrected VHM estimate. (Answer: $8.5$)
Three risk means are $10,14,20$, with total exposures $1,3,2$. The EPV estimate is $4$. Calculate VHM, retaining the exposure weights. (Answer: $208/11\approx18.9091$)

What You'll Learn