Basic Bootstrap Confidence Intervals

Why Reflect Bootstrap Errors?

A percentile bootstrap confidence interval uses quantiles of the bootstrap estimates themselves. A basic bootstrap interval starts from a different quantity: the bootstrap error, or the difference between a bootstrap estimate and the statistic calculated from the original sample. It uses the quantiles of those errors to estimate how far the original statistic may be from the population parameter.

The distinction matters when the bootstrap distribution is shifted relative to the observed statistic. A basic interval reflects the bootstrap quantiles around that statistic, rather than using the quantiles directly as endpoints. This can move both endpoints and can change the interval’s location. The method is a useful alternative, not a guarantee that the resulting interval is more accurate.

A bootstrap resample is drawn from the observed data using the resampling scheme appropriate to the study design, and the statistic is recalculated for that resample. Repeating this process produces bootstrap estimates. For this tutorial, assume the bootstrap quantiles are estimated from a sufficiently large collection of resamples; small numerical examples below use fewer values only to make the arithmetic visible.

The quantile convention can affect computed endpoints slightly, especially with a small number of replicates. In the first two examples, use the nearest-rank convention: sort the values and take rank \(\lceil Bp\rceil\) for quantile level \(p\), where \(B\) is the number of values. In practical work, use the stated quantile method consistently and report it when relevant.

How the Basic Interval Is Derived

The construction follows from treating the bootstrap errors as approximations to the sampling errors of the original statistic. The sampling error is \(\hat{\theta}-\theta\), where \(\theta\) is the population parameter. If the bootstrap-error quantiles approximate the corresponding sampling-error quantiles, a central \(1-\alpha\) range for the sampling error is approximately \(q^{*}_{\alpha/2}\) to \(q^{*}_{1-\alpha/2}\).

This gives the basic interval. The order reversal is essential: the upper error quantile is subtracted to get the lower endpoint, and the lower error quantile is subtracted to get the upper endpoint. In general, bootstrap errors do not have to be symmetric around zero.

Worked Construction and Comparison

The relationship to the percentile interval can be written directly. Since \(E^{*}=\hat{\theta}^{*}-\hat{\theta}\), subtracting a constant shifts every quantile by that constant: \(q^{*}_{p}=q_{p}(\hat{\theta}^{*})-\hat{\theta}\), using the same quantile convention. Substituting this into the basic-interval formula gives the equivalent expression below.

This is a reflection around \(\hat{\theta}\): each percentile endpoint is placed the same distance from the observed statistic on the opposite side. The basic interval need not have the same endpoints—or even the same width—as the percentile interval.

A Further Example: Reading Quantile Summaries

Assumptions and Practical Cautions

A basic interval relies on the bootstrap distribution being informative about the sampling error of the statistic. In an ordinary nonparametric bootstrap for independent observations, resampling is with replacement from the observed sample, which serves as an empirical approximation to the population. The observations must reasonably represent the population relevant to the question. If the design involves pairing, grouping, clustering, or another dependence structure, resampling individual observations independently may be inappropriate; the resampling scheme must preserve the structure that generated the data.

The statistic must also be calculable for the resamples, and the sample must provide useful information about its sampling behavior. A bootstrap procedure cannot repair a biased or unrepresentative sample simply by resampling it. Nor does reflecting quantiles automatically correct every form of bias, skewness, or instability. In particular, an interval can extend into impossible values for a parameter, or have an endpoint whose position is implausible for the context. Such outcomes are reasons to examine the method and data, not to silently alter the result.

The interval is an approximation, not a finite-sample guarantee. Its quality depends on the statistic, the data, the sampling design, and how well the bootstrap distribution represents sampling uncertainty. More bootstrap resamples can reduce simulation noise in estimated quantiles, but cannot eliminate the limitations caused by an unsuitable sampling model or inadequate original data.

Common Mistake: Reversing the Wrong Endpoints

A frequent error is to subtract the lower bootstrap-error quantile first and call the result the lower endpoint. That produces \(\hat{\theta}-q^{*}_{\alpha/2}\) as the lower limit, even though it is the larger of the two limits when the lower error quantile is below the upper one. The correction is to reverse the quantiles when subtracting: the lower interval endpoint is \(\hat{\theta}-q^{*}_{1-\alpha/2}\), and the upper endpoint is \(\hat{\theta}-q^{*}_{\alpha/2}\).

Another common mistake is to report the percentile interval after calculating bootstrap errors. The two procedures are related, but the basic method reflects the endpoints around the observed statistic. Check the result by confirming that the basic interval equals \([2\hat{\theta}-\text{percentile upper endpoint},\,2\hat{\theta}-\text{percentile lower endpoint}]\).