Why the Interval Method Matters
A bootstrap distribution can be skewed, shifted, or bounded even when the original estimate is a single number. Different confidence-interval methods use that distribution in different ways. Consequently, they can give noticeably different endpoints, particularly in small samples or when an estimator is sensitive to a few large claims. An interval should not be selected just because its calculation is familiar: its assumptions and endpoint behavior matter.
This tutorial compares normal, percentile, basic, studentized, and bias-corrected and accelerated (BCa) intervals. The goal is to understand what each method does with the bootstrap output and when its result deserves caution. The comparisons are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.
Five Ways to Turn Bootstrap Results Into an Interval
The normal and percentile methods are often the most direct. The basic method reflects the percentile interval around the observed estimate. The studentized method first puts estimation errors on a standard-error scale. BCa adjusts the quantile probabilities to account for estimated bias and acceleration. These methods do not become interchangeable merely because they use the same bootstrap replicates.
| Method | Interval construction | What it uses |
|---|---|---|
| Normal | \(\hat{\theta}\pm z_{1-\alpha/2}\widehat{\operatorname{SE}}_{\mathrm{boot}}\) | Estimate and bootstrap standard error |
| Percentile | \([q_{\alpha/2}^{*},q_{1-\alpha/2}^{*}]\) | Bootstrap estimate quantiles directly |
| Basic | \([2\hat{\theta}-q_{1-\alpha/2}^{*},\,2\hat{\theta}-q_{\alpha/2}^{*}]\) | Reflected bootstrap estimate quantiles |
| Studentized | \([\hat{\theta}-t_{1-\alpha/2}^{*}\widehat{\operatorname{SE}},\,\hat{\theta}-t_{\alpha/2}^{*}\widehat{\operatorname{SE}}]\) | Quantiles of replicate-specific standardized errors |
| BCa | \([q_{\alpha_{\mathrm{low}}}^{*},q_{\alpha_{\mathrm{high}}}^{*}]\) | Bootstrap quantiles at adjusted probabilities |
For the normal interval, \(\widehat{\operatorname{SE}}_{\mathrm{boot}}\) is the standard deviation of the bootstrap estimates. The method gives a symmetric interval around \(\hat{\theta}\). It is most credible when the estimator’s sampling distribution is approximately symmetric and normal, and its standard error is estimated adequately. Skewness can make symmetry inappropriate, while an unconstrained interval can cross a parameter boundary.
The percentile interval takes the lower and upper bootstrap estimate quantiles as its endpoints. It can reflect skewness and will respect a boundary if every bootstrap estimate respects that boundary. It relies on the bootstrap distribution being a good approximation to the estimator’s sampling distribution, including in its tails.
The basic interval reflects the percentile endpoints around \(\hat{\theta}\). It also uses bootstrap quantiles, but does not generally preserve a parameter boundary. The studentized method uses \(T^{*}=(\hat{\theta}^{*}-\hat{\theta})/\widehat{\operatorname{SE}}^{*}\), where \(\widehat{\operatorname{SE}}^{*}\) is an estimated standard error within that replicate. It can accommodate variation in the estimator’s standard error across resamples, but requires reliable replicate-specific standard errors and a useful approximation to the distribution of \(T^{*}\). BCa uses the bias-correction and acceleration quantities described below; it relies on a stable estimate of those adjustments and enough bootstrap replicates to resolve the adjusted tail probabilities.
Why the Basic Interval Reflects the Quantiles
The reflection in the basic interval follows by approximating the distribution of estimation errors. Let \(q_p(\hat{\theta}-\theta)\) denote the \(p\) quantile of the sampling error. Bootstrap errors are centered at the observed estimate, so their quantiles are \(q_p^{*}-\hat{\theta}\), not \(q_p^{*}\) itself. The bootstrap approximation is
An approximate confidence statement is that the error lies between its \(\alpha/2\) and \(1-\alpha/2\) quantiles. Substitute the bootstrap error quantiles and solve each inequality for \(\theta\):
The upper bootstrap quantile therefore becomes the lower basic endpoint. This reversal is a frequent source of mistakes.
Worked Comparisons
Worked Example: Comparing Four Intervals for a Skewed Estimate
A hypothetical portfolio analyst estimates a claim-cost parameter as \(\hat{\theta}=\$12.00\). A bootstrap run gives \(\widehat{\operatorname{SE}}_{\mathrm{boot}}=\$2.00\), estimate quantiles \(q_{0.05}^{*}=\$9.00\) and \(q_{0.95}^{*}=\$16.00\), and studentized-error quantiles \(t_{0.05}^{*}=-1.30\) and \(t_{0.95}^{*}=1.80\). Calculate four 90% intervals. Use \(z_{0.95}=1.6449\), and round dollar endpoints to the nearest cent.
Normal method. The margin is \(1.6449(\$2.00)=\$3.2898\). Thus the interval is \(\$12.00\pm \$3.2898\), or \([\$8.71,\$15.29]\) after rounding.
Percentile method. Use the supplied 5th and 95th bootstrap quantiles directly: \([\$9.00,\$16.00]\).
Basic method. Reflect the quantiles around \(\$12.00\). The lower endpoint is \(2(\$12.00)-\$16.00=\$8.00\); the upper endpoint is \(2(\$12.00)-\$9.00=\$15.00\). The interval is \([\$8.00,\$15.00]\).
Studentized method. Use the 95th studentized quantile for the lower endpoint and the 5th for the upper endpoint. The lower endpoint is \(\$12.00-1.80(\$2.00)=\$8.40\). The upper endpoint is \(\$12.00-(-1.30)(\$2.00)=\$14.60\). The interval is \([\$8.40,\$14.60]\).
The four results differ because the methods use different features of the bootstrap output. The mistake to avoid is treating the estimate quantiles as error quantiles in the basic method. For example, using \(\hat{\theta}-q_{0.95}^{*}\) and \(\hat{\theta}-q_{0.05}^{*}\) would give \([-\$4.00,\$3.00]\), which is wrong: the quantiles must first be centered by subtracting \(\hat{\theta}\), leading to the reflected endpoints \([\$8.00,\$15.00]\). This example’s arithmetic is checked by noting that the basic interval is centered at \((\$8+\$15)/2=\$11.50\), not necessarily at the estimate when the bootstrap quantiles are asymmetric around it.
Apply this adjustment separately to \(p=\alpha/2\) and \(p=1-\alpha/2\), then take the bootstrap estimate quantiles at those two adjusted probabilities. The adjusted probabilities need not be equally far from 0.5 and are generally not the nominal tail probabilities.
Worked Example: Calculating Adjusted BCa Tail Probabilities
A hypothetical 90% BCa interval has \(z_0=0.10\) and \(a=0\). The bootstrap estimate quantiles at probabilities 0.0743 and 0.9674 are \(\$8.10\) and \(\$17.60\), respectively. Find the adjusted probabilities and interval. Use \(z_{0.05}=-1.6449\), \(z_{0.95}=1.6449\), and round probabilities to four decimals.
For the lower endpoint, \(z_p=-1.6449\), so \(z_0+z_p=0.10-1.6449=-1.5449\). Because \(a=0\), the denominator is 1. The normal argument is therefore \(0.10-1.5449=-1.4449\), and \(\Phi(-1.4449)\approx0.0743\).
For the upper endpoint, \(z_p=1.6449\), so \(z_0+z_p=0.10+1.6449=1.7449\). Again, the denominator is 1. The normal argument is \(0.10+1.7449=1.8449\), and \(\Phi(1.8449)\approx0.9674\).
Thus the adjusted probabilities are 0.0743 and 0.9674. Taking the stated bootstrap quantiles at those probabilities gives the 90% BCa interval \([\$8.10,\$17.60]\). As a check, when \(a=0\), the formula reduces to \(\Phi(2z_0+z_p)\); the upper argument is \(0.20+1.6449=1.8449\), not \(1.7449\).
Worked Example: Checking Endpoint Behavior for a Positive Parameter
A hypothetical analyst estimates a positive tail-cost quantity as \(\hat{\theta}=\$4.00\), with bootstrap standard error \(\$3.00\). The bootstrap estimate quantiles for a 95% interval are \(q_{0.025}^{*}=\$1.20\) and \(q_{0.975}^{*}=\$10.50\). The studentized quantiles are \(t_{0.025}^{*}=-1.60\) and \(t_{0.975}^{*}=2.40\). A BCa calculation gives adjusted probabilities 0.010 and 0.990, with bootstrap quantiles \(\$0.70\) and \(\$13.00\). Compare the endpoint behavior, using \(z_{0.975}=1.96\).
The normal interval is \(\$4.00\pm1.96(\$3.00)=\$4.00\pm\$5.88\), which gives \([-\$1.88,\$9.88]\). The percentile interval is \([\$1.20,\$10.50]\), and the basic interval is \([2(\$4.00)-\$10.50,\,2(\$4.00)-\$1.20]=[-\$2.50,\$6.80]\).
For the studentized interval, the lower endpoint is \(\$4.00-2.40(\$3.00)=-\$3.20\); the upper endpoint is \(\$4.00-(-1.60)(\$3.00)=\$8.80\). The BCa interval uses the supplied adjusted quantiles directly, giving \([\$0.70,\$13.00]\).
The normal, basic, and studentized intervals have negative lower endpoints, even though the parameter is positive. The percentile and BCa intervals respect positivity here because the bootstrap estimates and selected quantiles are positive; that is not a universal guarantee of accurate coverage. Do not silently replace negative endpoints by zero: truncation changes the procedure and can change its coverage. The calculation is checked by confirming that each interval’s endpoints use the correct tail quantiles, with the upper studentized quantile used for the lower endpoint.
Choosing and Interpreting an Interval
Skewness alone does not determine a universally best method. Consider the estimator, its bootstrap distribution, the quality of the standard-error estimates, and any parameter constraints. The percentile method is a direct choice when the bootstrap distribution adequately represents the estimator’s distribution and its quantiles are stable. The basic method can be useful when an error-centered interpretation is appropriate, but reflection can produce implausible endpoints. The normal method is simple but can conceal asymmetry. Studentization can adjust for changing variability, at the cost of needing reliable standard errors in every replicate. BCa adjusts both bias and asymmetry-related effects, but its tail probabilities and endpoints can be unstable if the bootstrap run is too small or the adjustment is extreme.
A practical comparison should examine the bootstrap distribution, the interval endpoints, and the assumptions behind each method. In particular, check whether a parameter boundary is relevant and whether a reported endpoint violates it. Also check that the bootstrap replicates are independent resamples generated under the intended resampling scheme; a coding or resampling error cannot be repaired by choosing a different interval formula.
- Do not confuse estimate quantiles with error quantiles. Error quantiles are \(q_p^{*}-\hat{\theta}\).
- Do not swap the basic or studentized tails. The upper error quantile determines the lower confidence endpoint.
- Do not assume a symmetric interval is appropriate just because it is convenient. Compare it with the bootstrap distribution’s shape.
- Do not treat a boundary violation as harmless. It may indicate a poor approximation or a mismatch between the interval method and the parameter space.
Check Your Understanding
For each problem, calculate the requested interval or identify the relevant limitation. Answers are rounded as stated.
- A bootstrap estimate is \(\hat{\theta}=25\), with bootstrap standard error 4. Calculate the 90% normal interval using \(z_{0.95}=1.6449\). Round to two decimals. (Answer: \([18.42,31.58]\).)
- For \(\hat{\theta}=21\), the 2.5th and 97.5th bootstrap estimate quantiles are 14 and 29. Give the 95% percentile and basic intervals. (Answer: Percentile \([14,29]\); basic \([13,28]\).)
- For a 95% studentized interval, \(\hat{\theta}=8\), \(\widehat{\operatorname{SE}}=1.5\), \(t_{0.025}^{*}=-2.1\), and \(t_{0.975}^{*}=1.7\). Calculate the endpoints. (Answer: \([5.45,11.15]\).)
- A 90% BCa interval has \(z_0=0.10\) and \(a=0\). What adjusted tail probabilities should be used? (Answer: Approximately 0.0743 and 0.9674.)
- A positive parameter has estimate 3.0 and bootstrap standard error 2.0. A 95% normal interval uses \(z_{0.975}=1.96\). Does its lower endpoint respect the parameter boundary, and what is the endpoint? (Answer: No; \(3.0-1.96(2.0)=-0.92\).)