BCa Bootstrap Intervals

Why Adjust a Bootstrap Interval?

A bootstrap interval is built from repeated estimates calculated on resampled data. The Normal and Basic Bootstrap Intervals use the bootstrap distribution in different ways, but both can be less accurate when that distribution is biased relative to the observed estimate or is asymmetric. A bias-corrected and accelerated (BCa) interval adjusts the interval's tail probabilities to account for both features.

BCa intervals are especially useful when the statistic's sampling distribution is skewed or when its bias is not negligible. They do not eliminate every source of uncertainty: the quality of the interval still depends on whether the resampling procedure represents the data-generating process and whether the statistic is sufficiently regular for bootstrap approximation to work well.

The usual setup assumes independent observations sampled from a common population and bootstrap samples drawn with replacement from the empirical distribution of the observed data. The statistic is recalculated in each bootstrap sample using the same definition as for the original estimate. If observations are dependent, clustered, or sampled under a complex design, the resampling scheme must reflect that structure; the ordinary observation-level bootstrap may not be appropriate.

The Two Corrections

The bias correction measures where the original estimate falls in the bootstrap distribution. Define

where \(\Phi\) is the standard normal cumulative distribution function. Thus, if about half the bootstrap estimates are below \(\hat{\theta}\), then \(z_0\) is near zero. If substantially more than half are below it, \(z_0\) is positive, indicating that the original estimate lies relatively high in the bootstrap distribution.

The strict inequality in this definition is one common convention. When bootstrap estimates tie the original estimate, software and texts may use different tie conventions, so the convention should be stated and used consistently. If \(\hat{q}=0\) or \(1\), \(z_0\) is infinite. A finite-sample implementation may clip \(\hat{q}\) to an interior value, such as the interval from \(1/(2B)\) to \(1-1/(2B)\), but such clipping does not create information about unobserved tails.

The acceleration measures how the statistic's variability changes across the data. For each original observation \(i\), calculate the leave-one-out estimate \(\hat{\theta}_{(i)}\), using all observations except \(i\). Let \(\bar{\theta}_{(.)}\) be the average of these \(n\) jackknife estimates. Then

The jackknife is used here to estimate acceleration; it is not a second bootstrap interval. If every leave-one-out estimate is identical, both sums in the formula are zero and the formula is undefined. That case requires further judgment about the statistic and data, rather than automatic substitution into the formula.

How the Adjusted Endpoints Arise

For a two-sided interval with nominal confidence level \(1-\alpha\), define the nominal normal scores \(z_{\alpha/2}=\Phi^{-1}(\alpha/2)\) and \(z_{1-\alpha/2}=\Phi^{-1}(1-\alpha/2)\). The BCa adjusted probabilities are

The endpoints are the \(p_L\) and \(p_U\) quantiles of the bootstrap estimates. This is a quantile interval: it is not formed by adding and subtracting a standard error from \(\hat{\theta}\). For a finite set of replicates, a quantile convention is needed. In the examples here, use type-7 linear interpolation: with sorted estimates \(\hat{\theta}^{*}_{(1)}\le\cdots\le\hat{\theta}^{*}_{(B)}\), the position for probability \(p\) is \(h=1+(B-1)p\). Interpolate between the two adjacent ordered estimates if \(h\) is not an integer.

The adjustment can be understood by expressing the bootstrap distribution on a normal-score scale. The bias correction shifts the score origin by \(z_0\). Acceleration then modifies a score \(z\) through the transformation \(z\mapsto z_0+(z_0+z)/[1-a(z_0+z)]\). Applying \(\Phi\) converts that adjusted score back to a probability, which explains why the final endpoints are quantiles at adjusted probabilities.

Why the Acceleration Formula Uses Jackknife Deviations

A short justification clarifies the role of the cubic and squared deviations. Write \(\delta_i=\bar{\theta}_{(.)}-\hat{\theta}_{(i)}\). The jackknife deviations sum to zero, and their third-power sum records asymmetry: positive and negative deviations contribute with opposite signs. Their squared sum measures overall variation in the leave-one-out estimates. The acceleration is the standardized third-power sum divided by six:

This form arises from the local change in the statistic's standard error as the underlying distribution is perturbed. A symmetric pattern of jackknife deviations has a zero cubic sum and hence \(a=0\); an asymmetric pattern produces nonzero acceleration. The factor \(1/6\) is part of the BCa adjustment, not an optional scaling. The jackknife formula is an approximation to this local sensitivity, so it can be unreliable for highly nonsmooth statistics or very small samples.

Common Traps and Practical Interpretation

The calculation has several places where a small procedural error can change an endpoint. Check the following before reporting a BCa interval:

• Count relative to the original estimate. \(z_0\) uses the proportion of bootstrap estimates below \(\hat{\theta}\), not the proportion below a nominal percentile or the fraction above \(\hat{\theta}\).
• Preserve the sign of acceleration. The cubic deviations are signed. Squaring them, taking absolute values, or reversing their direction without changing the formula gives an incorrect acceleration.
• Use the correct normal scores. A two-sided \(1-\alpha\) interval uses \(z_{\alpha/2}\) and \(z_{1-\alpha/2}\); a one-sided confidence bound uses different nominal tail probabilities.
• Do not confuse adjusted probabilities with normal scores. The expression inside \(\Phi\) is a score; the result after applying \(\Phi\) is the probability used to select a bootstrap quantile.
• State the quantile rule. Different software may interpolate differently. For type 7, the position is \(1+(B-1)p\), not \(Bp\). With \(B=6\) and \(p=0.80\), for example, \(h=1+5(0.80)=5\), so the endpoint is the fifth ordered estimate, not an interpolation beyond it.

BCa intervals matter because an apparently reasonable symmetric interval can miss important asymmetry or bias in the bootstrap distribution. At the same time, extreme adjusted probabilities can require extrapolation beyond the range of a finite set of replicates in other quantile conventions, and a small bootstrap sample may provide a coarse estimate of a tail. A reported BCa interval should therefore include enough information to identify the confidence level, the resampling scheme, and the quantile convention.