BCa Intervals for Bias and Skewness

Why Adjust Bootstrap Percentiles?

A percentile bootstrap interval uses selected quantiles of the bootstrap estimates. It is simple to construct, but its tail cutoffs can be poorly placed when the bootstrap distribution is shifted relative to the estimate from the original sample or when the estimator’s sampling behavior is skewed. The bias-corrected and accelerated interval, usually abbreviated BCa, adjusts the percentile levels to address these two features.

BCa does not change the bootstrap estimates themselves. It changes which quantiles of their distribution are used as the interval endpoints. Consequently, a BCa interval can be asymmetric around the observed estimate, and its lower and upper tails can have different adjusted probabilities.

Definition: For an estimate \(\hat{\theta}\), a BCa interval adjusts the nominal lower and upper tail probabilities using (1) a bias-correction term \(z_0\), calculated from the position of \(\hat{\theta}\) within the bootstrap distribution, and (2) an acceleration term \(a\), calculated from delete-one jackknife estimates. The endpoints are the bootstrap quantiles at the adjusted probabilities.

The Two Adjustments

Suppose the original sample contains \(n\) observational units and produces estimate \(\hat{\theta}\). Resample those units with replacement to obtain \(B\) bootstrap samples, and calculate the estimate in each. Write the resulting bootstrap estimates in increasing order as \(\hat{\theta}^{*}_1,\ldots,\hat{\theta}^{*}_B\). This construction assumes that the observations are sampled independently from the population and that resampling whole observations is appropriate for the study design. The bootstrap procedure must preserve any relevant structure in the data.

The bias-correction term measures where the original estimate sits relative to the bootstrap estimates. Let \(r\) be the number of bootstrap estimates below \(\hat{\theta}\), and let \(t\) be the number tied with it. Using half weight for ties, define the estimated fraction below the original estimate as \(q=(r+t/2)/B\). Then

z_0=\Phi^{-1}(q),

where \(\Phi\) is the standard normal cumulative distribution function. Thus, \(z_0=0\) when half the bootstrap estimates are below \(\hat{\theta}\), approximately. A positive \(z_0\) means the original estimate is relatively high in the bootstrap distribution; a negative \(z_0\) means it is relatively low. This is a correction on the normal-score scale, not an estimate of bias in the units of \(\theta\).

The acceleration term describes how the estimate’s variability changes across the data values. To calculate it, delete one observational unit at a time and recompute the estimate. Denote the estimate with observation \(i\) deleted by \(\hat{\theta}_{(i)}\), and let \(\bar{\theta}_{(.)}\) be the mean of these \(n\) jackknife estimates. The acceleration is

a= \frac{\displaystyle\sum_{i=1}^{n} \left(\bar{\theta}_{(.)}-\hat{\theta}_{(i)}\right)^3} {\displaystyle 6\left[ \sum_{i=1}^{n}\left(\bar{\theta}_{(.)}-\hat{\theta}_{(i)}\right)^2 \right]^{3/2}}.

The cubic deviations in the numerator make the sign and size of \(a\) sensitive to asymmetry among the jackknife estimates. The squared deviations in the denominator scale the result. Acceleration is dimensionless; it is not measured in the units of the estimate.

Adjusted Probabilities and BCa Endpoints

For a two-sided interval with nominal confidence level \(1-\alpha\), let \(z_{\alpha/2}\) and \(z_{1-\alpha/2}\) be the corresponding standard normal quantiles. Transform each normal score \(z\) into an adjusted probability:

p(z)=\Phi\!\left[ z_0+\frac{z_0+z}{1-a(z_0+z)} \right].

The adjusted lower and upper probabilities are \(p_L=p(z_{\alpha/2})\) and \(p_U=p(z_{1-\alpha/2})\). The BCa interval is the interval between the bootstrap quantiles at \(p_L\) and \(p_U\). In a finite set of \(B\) replicates, software may use an interpolation rule to obtain a quantile between ordered estimates; the rule should be stated when exact reproducibility matters.

Key result: For a 95% BCa interval, transform the standard normal scores \(-1.96\) and \(1.96\) using \(z_0\) and \(a\). The resulting probabilities are generally not 0.025 and 0.975. Take the corresponding quantiles of the bootstrap estimates as the lower and upper endpoints.

Why the Transformation Has These Features

The transformation can be understood by checking what happens when neither adjustment is needed. If \(z_0=0\) and \(a=0\), then

p(z)=\Phi\!\left(0+\frac{0+z}{1-0}\right)=\Phi(z).

The nominal normal scores therefore return the ordinary percentile probabilities. A nonzero \(z_0\) shifts the score to account for the observed estimate’s position in the bootstrap distribution. The acceleration term changes the denominator according to the shifted score, so the lower and upper tails need not move by equal amounts. This score-scale adjustment is the reason a BCa interval can accommodate both relative bias and asymmetry.

This explanation verifies the basic calibration and role of each term; it does not guarantee that the bootstrap approximates the sampling distribution well for every statistic or data set. The method is most defensible when the estimator is sufficiently regular and the observations provide a reasonable basis for resampling. If the denominator \(1-a(z_0+z)\) is zero or very close to zero, the transformation is unstable and the resulting adjusted probability may be unusable. Such a result is a warning, not a reason to force a conventional interval.

Worked Example: Calculating Both Adjustments

In a hypothetical analysis, the original estimate of a typical completion time is \(\hat{\theta}=6.0\) minutes. There are \(B=100\) bootstrap replicates, of which 55 are below 6.0 minutes and none are tied. The three delete-one jackknife estimates are 4.0, 7.0, and 7.0 minutes. Calculate \(z_0\), \(a\), and the adjusted probabilities for a 95% BCa interval.

Bias correction. The fraction below the original estimate is \(q=55/100=0.55\). Since \(\Phi^{-1}(0.55)\approx0.1257\),

z_0=\Phi^{-1}(0.55)\approx0.1257.

This positive value says the original estimate is toward the high side of the bootstrap distribution. It does not mean the estimated bias is 0.1257 minutes.

Acceleration. The mean of the jackknife estimates is \(\bar{\theta}_{(.)}=(4+7+7)/3=6\) minutes. The deviations \(\bar{\theta}_{(.)}-\hat{\theta}_{(i)}\) are 2, \(-1\), and \(-1\) minutes. Therefore,

a= \frac{2^3+(-1)^3+(-1)^3} {6\left[2^2+(-1)^2+(-1)^2\right]^{3/2}} = \frac{6}{6(6)^{3/2}} \approx0.0680.

Adjusted probabilities. For the lower 95% limit, use \(z=-1.96\). Then \(z_0+z=0.1257-1.96=-1.8343\), and

p_L= \Phi\!\left(0.1257+ \frac{-1.8343}{1-0.0680(-1.8343)}\right) = \Phi\!\left(0.1257+\frac{-1.8343}{1.1247324}\right) \approx\Phi(-1.5052) \approx0.0661.

For the upper limit, use \(z=1.96\). The transformed score is

0.1257+ \frac{0.1257+1.96}{1-0.0680(0.1257+1.96)} = 0.1257+\frac{2.0857}{0.8581724} \approx2.5561,

so \(p_U\approx\Phi(2.5561)\approx0.9947\). With 100 ordered bootstrap estimates and a nearest-rank convention, these probabilities select approximately the 7th and 100th order statistics. For example, if those values are 5.1 and 10.2 minutes, the BCa interval is approximately 5.1 to 10.2 minutes. The adjusted tail probabilities are noticeably different from 0.025 and 0.975, so the interval need not be centered on 6.0 minutes.

Worked Example: Bias Correction With No Acceleration

A second hypothetical analysis has \(B=200\) bootstrap replicates. Of these, 130 are below the original estimate and none are tied. Suppose the jackknife estimates give \(a=0\). Find the adjusted probabilities for a 95% BCa interval.

The proportion below the original estimate is \(130/200=0.65\), giving \(z_0=\Phi^{-1}(0.65)\approx0.3853\). With \(a=0\), the transformation reduces to \(p(z)=\Phi(2z_0+z)\). The adjusted lower score is \(-1.96+2(0.3853)=-1.1894\), and the adjusted upper score is \(1.96+2(0.3853)=2.7306\). Thus,

p_L=\Phi(-1.1894)\approx0.1171, \qquad p_U=\Phi(2.7306)\approx0.9968.

The interval uses approximately the 11.71st and 99.68th percentiles of the bootstrap estimates. The unequal adjustments arise here from the bias correction alone; zero acceleration does not undo a nonzero \(z_0\). The actual interval endpoints still depend on the bootstrap estimates and their quantile rule.

Worked Example: Acceleration Alters the Two Tails Differently

In a third hypothetical analysis, suppose the bias-correction term is \(z_0=0\), while three jackknife deviations \(\bar{\theta}_{(.)}-\hat{\theta}_{(i)}\) are \(-2\), 1, and 1 in the estimate’s units. Calculate the acceleration and the adjusted probabilities for a 95% interval.

The deviations sum to zero, consistent with being deviations from their mean. Their cubes sum to \(-8+1+1=-6\), and their squares sum to \(4+1+1=6\). Hence,

a=\frac{-6}{6(6)^{3/2}}\approx-0.0680.

For either tail, the denominator is \(1-(-0.0680)z\). At \(z=-1.96\), it is \(1-0.13328=0.86672\), giving a transformed score of \(-1.96/0.86672\approx-2.2614\) and \(p_L\approx0.0119\). At \(z=1.96\), the denominator is \(1-(-0.0680)(1.96)=1.13328\), giving a score of \(1.96/1.13328\approx1.7295\) and \(p_U\approx0.9581\). The different denominator values show why acceleration can move the tails unequally even when \(z_0=0\). These adjusted probabilities should be checked for sensible ordering and stability before using them to select endpoints.

Interpreting and Checking a BCa Interval

Suppose a BCa interval for a population median completion time is 5.1 to 10.2 minutes. A suitable interpretation is: using this bootstrap procedure, the 95% BCa confidence interval for the population median completion time extends from 5.1 to 10.2 minutes. The interval’s width and asymmetry describe the uncertainty represented by this method. They do not establish that there is a 95% probability that the fixed population median lies inside this particular interval.

Before reporting a BCa interval, check the foundations of the calculation:

The resampling unit matches the independent observational unit and preserves the design.
The estimate is calculated consistently from the original sample, every bootstrap sample, and every jackknife sample.
The bootstrap replicate count is large enough to represent the adjusted tail quantiles adequately.
The adjusted probabilities lie between 0 and 1, and the transformed denominators are not near zero.
The endpoints are reported in the parameter’s units and are ordered from lower to upper.

BCa intervals are not a remedy for every limitation of ordinary resampling. A small sample, an estimator with poor behavior under resampling, or data containing influential observations can make the bootstrap approximation unreliable. A numerical interval can be calculated even when its assumptions are not convincing, so the design and data quality still matter.

Common Mistake

A common mistake is to calculate \(z_0\) and \(a\), then report the ordinary 2.5th and 97.5th bootstrap percentiles anyway. That ignores the point of the adjustment. Another is to describe \(z_0\) as the bias in the estimate’s original units. In fact, \(z_0\) is a standard-normal score derived from the bootstrap rank of the original estimate. Calculate \(p_L\) and \(p_U\) with the transformation first, then use those probabilities to obtain the endpoints and interpret the interval in context.

Key takeaway: BCa uses the bootstrap rank of the original estimate to correct for relative bias and delete-one jackknife estimates to calculate acceleration. These adjustments change the quantile levels, so BCa endpoints may be asymmetric and should be interpreted as confidence limits in the parameter’s units.

Check Your Understanding

Try these questions before reviewing the explanations above.

If 40% of bootstrap estimates fall below the original estimate and there are no ties, is \(z_0\) positive, negative, or zero? What does its sign indicate?
For jackknife estimates 8, 10, and 12, calculate their mean and the three deviations used in the acceleration formula.
If \(z_0=0\) and \(a=0\), what probabilities are used for a nominal 95% BCa interval?
Why can the lower and upper BCa tail probabilities differ by unequal amounts from their ordinary percentile levels?
State a correct interpretation of a hypothetical 95% BCa interval of 4.2 to 7.8 hours for a population mean duration.