Bootstrap Standard Errors For Frequency And Severity Estimates

Why Bootstrap Standard Errors Are Useful

A frequency or severity estimate is calculated from a finite sample, so it would generally change if a different sample were observed. A bootstrap standard error estimates the resulting sampling variability by repeatedly resampling the observed data and recalculating the estimator. The essential step is to recompute the estimate in every replicate; a standard error cannot be found merely by inspecting the original data or by resampling without regard to what the estimator uses.

For frequency, the sampling unit is typically an exposure record, such as a policy-period, with its claim count and exposure. For severity, the sampling units are typically individual observed claims. These choices reflect different data structures: a claim-level sample contains information about the sizes of observed claims, but by itself does not represent the number of claim-free policies or the exposure underlying a frequency rate.

Definition: Let $\hat{\theta}$ be an estimator calculated from the observed data. For each of $B$ bootstrap replicates, resample the appropriate observational units and calculate $\hat{\theta}^{*(b)}$, where $b=1,\ldots,B$. The bootstrap mean and bootstrap standard error are $$ \bar{\theta}^{*}=\frac{1}{B}\sum_{b=1}^{B}\hat{\theta}^{*(b)}, \qquad \widehat{\operatorname{SE}}_{\mathrm{boot}}(\hat{\theta}) =\sqrt{\frac{1}{B-1}\sum_{b=1}^{B}\left(\hat{\theta}^{*(b)}-\bar{\theta}^{*}\right)^2}. $$ The bootstrap standard error is the sample standard deviation of the replicate estimates. It estimates the sampling variability of $\hat{\theta}$, using the empirical distribution of the observed sampling units as an approximation to the population distribution.

The $B-1$ denominator is used because the replicate estimates are being treated as a sample from the bootstrap distribution, and their mean $\bar{\theta}^{*}$ is estimated from those same $B$ values. This is the usual sample-variance calculation. Increasing $B$ usually reduces the simulation noise in the estimated standard error, but it does not fix an inappropriate resampling unit or a data sample that poorly represents the population.

Choose the Sampling Unit Before Calculating

For $n$ policy-periods with equal exposure, let $N_j$ be the claim count for policy-period $j$. The observed estimate of mean claim frequency per policy-period is $$ \hat{\lambda}=\frac{1}{n}\sum_{j=1}^{n}N_j. $$ A bootstrap replicate resamples $n$ policy-period records with replacement and recalculates this mean count. For unequal exposures $e_j$, a common rate estimate is $$ \hat{\lambda}=\frac{\sum_{j=1}^{n}N_j}{\sum_{j=1}^{n}e_j}. $$ Each replicate must resample policy-period records together with their exposure, then recompute the ratio of total claims to total exposure. Resampling counts while leaving an unrelated denominator fixed can produce a different estimator.

For observed claim severities $X_1,\ldots,X_m$, the sample mean severity estimate is $$ \hat{\mu}=\frac{1}{m}\sum_{j=1}^{m}X_j. $$ A severity bootstrap resamples $m$ claims with replacement and recalculates the mean in each replicate. The same recompute-and-compare procedure applies to other specified frequency or severity estimators, but the estimator must be evaluated in precisely the same way for every replicate as it is for the observed data.

Worked Example: Standard Error of an Equal-Exposure Frequency Estimate

A hypothetical insurer observes five one-year policy-periods with claim counts $0,0,1,1,2$, each with exposure $1$. Estimate the bootstrap standard error of the mean claim count per policy-period using the four bootstrap samples listed below. No interest rate or mortality model is involved; the bootstrap distribution is based on the empirical distribution of the five observed policy-period records.

Method and observed estimate. The original total is $0+0+1+1+2=4$, so the observed frequency estimate is $\hat{\lambda}=4/5=0.8$ claims per policy-period. Each replicate also contains five policy-period records, so its estimate is its total count divided by $5$.

Recompute each replicate. The sampled claim counts and estimates are:

Replicate 1: $0,0,0,1,1$. Total $=2$, so $\hat{\lambda}^{*(1)}=2/5=0.4$.
Replicate 2: $0,0,1,1,1$. Total $=3$, so $\hat{\lambda}^{*(2)}=3/5=0.6$.
Replicate 3: $0,1,1,1,1$. Total $=4$, so $\hat{\lambda}^{*(3)}=4/5=0.8$.
Replicate 4: $0,1,1,1,2$. Total $=5$, so $\hat{\lambda}^{*(4)}=5/5=1.0$.

The mean of these four replicate estimates is $\bar{\lambda}^{*}=(0.4+0.6+0.8+1.0)/4=0.7$. Their deviations from $0.7$ are $-0.3,-0.1,0.1,0.3$; the squared deviations are $0.09,0.01,0.01,0.09$, with sum $0.20$. Thus the replicate sample variance is $0.20/(4-1)=0.0666667$, and the bootstrap standard error is $\sqrt{0.0666667}=0.2581989$. To four decimal places, it is 0.2582 claims per policy-period.

Check and common mistake. The four replicate estimates are centered at $0.7$, not at the observed estimate $0.8$; the standard deviation formula uses their replicate mean. This small set of only four replicates illustrates the calculation, but in practice a much larger $B$ is needed for a stable estimate.

Worked Example: Standard Error of Mean Claim Severity

A hypothetical portfolio has four observed claim severities of $\$100,\$200,\$300,\$400$. Estimate the bootstrap standard error of the sample mean using the four listed claim-level resamples, each of size four. No interest rate or mortality assumption is relevant; the resampling distribution is the empirical distribution assigning probability $1/4$ to each observed severity.

Method and original estimate. The original total severity is $100+200+300+400=\$1{,}000$, so the sample mean is $\hat{\mu}=\$1{,}000/4=\$250$.

Recompute each replicate mean.

Replicate 1: $\$100,\$100,\$200,\$200$. The total is $\$600$, giving $\hat{\mu}^{*(1)}=\$150$.
Replicate 2: $\$100,\$200,\$300,\$400$. The total is $\$1{,}000$, giving $\hat{\mu}^{*(2)}=\$250$.
Replicate 3: $\$200,\$300,\$300,\$400$. The total is $\$1{,}200$, giving $\hat{\mu}^{*(3)}=\$300$.
Replicate 4: $\$100,\$300,\$400,\$400$. The total is $\$1{,}200$, giving $\hat{\mu}^{*(4)}=\$300$.

The bootstrap mean is $\bar{\mu}^{*}=(150+250+300+300)/4=\$250$. The deviations are $-\$100,\$0,\$50,\$50$, and the squared deviations are $10{,}000,0,2{,}500,2{,}500$, in dollars squared. Their sum is $15{,}000$, so the replicate sample variance is $15{,}000/3=5{,}000$ dollars squared. The bootstrap standard error is $\sqrt{5{,}000}=\$70.7107$, or \$70.71 to the nearest cent.

Check and common mistake. The standard error describes the variability of the estimated mean across resamples; it is not the standard deviation of the four original claim amounts. Also, the four displayed replicates are only a specified subset of possible bootstrap samples, so their standard deviation is a demonstration of the calculation, not a claim that four replicates will give a precise practical estimate.

Unequal Exposure Requires a Recomputed Rate

When policy-periods have different exposure, the rate estimator is a ratio. In every bootstrap replicate, both the claim-count total and exposure total can change, because the policy-period record is resampled as a unit. The replicate rate is therefore not generally the average of the individual policy rates, and it is not the replicate claim total divided by the original portfolio exposure.

Worked Example: Frequency Standard Error With Unequal Exposures

A hypothetical book contains four policy-period records $(N_j,e_j)$: $(0,1),(2,2),(1,1),(3,2)$, where $N_j$ is claim count and $e_j$ is exposure in policy-years. Use the four bootstrap samples below to estimate the standard error of the total-claims-to-total-exposure rate. No interest rate or mortality model is used; resampling is from the empirical distribution of these four paired records.

Observed estimate. The total claim count is $0+2+1+3=6$, and total exposure is $1+2+1+2=6$ policy-years. Thus $\hat{\lambda}=6/6=1.0$ claim per policy-year.

Recompute the numerator and denominator in each replicate.

Replicate 1 contains all four original records: total claims $=6$, exposure $=6$, rate $=6/6=1.000000$.
Replicate 2 contains $(0,1),(0,1),(2,2),(1,1)$: total claims $=3$, exposure $=5$, rate $=3/5=0.600000$.
Replicate 3 contains $(2,2),(2,2),(1,1),(3,2)$: total claims $=8$, exposure $=7$, rate $=8/7=1.142857$.
Replicate 4 contains $(0,1),(1,1),(3,2),(3,2)$: total claims $=7$, exposure $=6$, rate $=7/6=1.166667$.

The mean replicate rate is $(1+0.6+1.142857+1.166667)/4=0.977381$. The squared deviations from that mean, rounded here for display, are $0.000512,\ 0.142416,\ 0.027382,\ 0.035829$. Their sum is $0.206139$; dividing by $4-1$ gives a sample variance of $0.068713$. Its square root is $0.2621$. The estimated bootstrap standard error is therefore 0.2621 claims per policy-year, to four decimal places.

Check and common mistake. Each replicate's exposure is the sum of the exposures in that replicate: for example, replicate 2 has exposure $1+1+2+1=5$. Keeping the original denominator of $6$ would incorrectly give $3/6$ rather than the replicate rate $3/5$, and would not reproduce the stated estimator.

Why the Replicate Standard Deviation Estimates Variability

The bootstrap treats the observed sampling units as an empirical population. Conditional on the data, each replicate is generated by drawing from that empirical population with replacement and applying the estimator to the resulting sample. Consequently, the replicate estimates behave as draws from an approximate sampling distribution for the estimator.

The standard deviation of that bootstrap distribution is its standard error. With a finite number $B$ of simulated replicates, the calculation uses the sample standard deviation instead. In particular, if the replicate estimates are independent draws from the bootstrap distribution and that distribution has finite variance, the sample variance with denominator $B-1$ is an unbiased estimate of the conditional bootstrap variance. Taking its square root gives the usual bootstrap standard error estimate.

This justifies the calculation, but it does not make the estimate exact. The bootstrap approximates population sampling behavior through the observed empirical distribution. If the sample misses important features of the population, or if the sampling units are not independent in the way the resampling assumes, the resulting standard error can be misleading. The bootstrap standard error also measures sampling variability, not bias, and it is not itself a confidence interval.

Common Exam Traps and Interpretation

Resampling the wrong units. Resampling claims alone cannot capture uncertainty in the number of claims per exposure when the frequency data are policy-period counts. Use the units that contain the information required by the estimator.
Failing to recalculate the estimator. For an exposure-adjusted rate, calculate each replicate's claims divided by that replicate's exposure. For severity, calculate the selected severity statistic anew in each replicate.
Using $B$ instead of $B-1$. The standard bootstrap standard error is the sample standard deviation of the replicate estimates, so the variance denominator is $B-1$.
Confusing claim dispersion with estimator uncertainty. The variability of individual claim amounts and the variability of a sample mean are different quantities. The bootstrap standard error is calculated from replicate estimates, not by reporting the standard deviation of individual observations.
Overinterpreting a small number of replicates. A small $B$ can make the estimated standard error unstable. More replicates reduce Monte Carlo error, but do not repair biased sampling or a poorly chosen resampling unit.

In practice, report the estimator, the sampling unit, the number $B$ of bootstrap replicates, and the bootstrap standard error with appropriate units. For example, a frequency standard error is expressed in claims per exposure unit, while a severity standard error is expressed in dollars. A clear interpretation is that resampling estimates how much the chosen estimator would vary under repeated sampling represented by the observed data—not that the parameter itself is random.

Key takeaway: A bootstrap standard error is the sample standard deviation of estimates recomputed across bootstrap replicates. Resample the correct observational units, recalculate the complete estimator each time, and interpret the result as an estimate of sampling variability.

Check Your Understanding

For each question, use the sample standard deviation of the bootstrap replicate estimates, with denominator $B-1$. Give money amounts to the nearest cent and rates to four decimal places where appropriate.

A hypothetical insurer has five equal-exposure policy-periods. Four bootstrap samples each have five records and produce claim totals $2,3,4,5$. Calculate the bootstrap standard error of the estimated claim frequency per policy-period. (Answer: $0.2582$ claims per policy-period)
Four claim-level bootstrap samples produce mean severities of $\$120,\$150,\$180,\$210$. Calculate the bootstrap standard error of the mean severity. (Answer: $\$38.73$)
Five exposure-adjusted frequency replicates have claim-count and exposure totals $(2,4),(3,4),(4,4),(5,4),(6,4)$. Recalculate the five rates and find their bootstrap standard error. (Answer: $0.3953$ claims per exposure unit)
A candidate calculates the standard error from four replicate frequency estimates $2.0,2.5,3.0,3.5$ by dividing the sum of squared deviations from their mean by $4$, then taking the square root. Identify the denominator error and calculate the standard bootstrap standard error. (Answer: Use $B-1=3$, not $B=4$; the standard error is $0.6455$)
Four bootstrap replicates give frequency estimates $0.5,1.0,1.5,2.0$ claims per policy-year and severity estimates $\$80,\$100,\$120,\$140$. Calculate both bootstrap standard errors and state their units. (Answer: Frequency $0.6455$ claims per policy-year; severity $\$25.82$)

What You'll Learn