BCa Bootstrap (Bias-Corrected and Accelerated)
Bias-Corrected and Accelerated Bootstrap · Also known as: BCa Bootstrap (Bias-Corrected Accelerated), bias-corrected accelerated bootstrap, BCa confidence interval
The BCa bootstrap is a resampling method, introduced by Bradley Efron in 1987, that produces more accurate confidence intervals than the plain percentile bootstrap by applying a bias correction and an acceleration adjustment. It is recommended for skewed distributions and small samples.
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When to use it
Use BCa when you need an accurate confidence interval for a statistic whose sampling distribution may be skewed or whose estimator is biased, especially with small to moderate samples (the method needs at least about 15 observations). It works for continuous or categorical outcomes and assumes observations are independent. Because acceleration is estimated by jackknife, BCa needs enough data for that estimate to be stable; with fewer than 20 observations the plain bootstrap is safer, and below about 10 a permutation test is preferable.
Strengths & limitations
- Second-order accurate: corrects both the bias and the skewness of the bootstrap distribution, so coverage is closer to nominal than the percentile method.
- Transformation-respecting and range-preserving, so endpoints stay within valid bounds for the statistic.
- Works without a normality assumption and is well suited to skewed distributions and small to moderate samples.
- Requires many resamples (B ≥ 2000 recommended), so it is computationally heavier than a single percentile interval.
- The acceleration constant relies on a jackknife estimate that becomes unreliable in very small samples (n < 20).
- Below about 10 observations the bias and acceleration corrections lose meaning and the method can mislead.
Frequently asked
How is BCa different from the percentile bootstrap?
The percentile method reads the interval directly off the bootstrap distribution's quantiles, which is accurate only if that distribution is unbiased and symmetric. BCa adjusts the quantiles with a bias-correction term and an acceleration term, giving second-order accurate coverage even when the estimator is biased or the distribution is skewed.
What do the bias-correction and acceleration constants do?
The bias-correction constant ẑ₀ measures how far the bootstrap distribution is shifted relative to the original estimate. The acceleration constant a, estimated by jackknife, captures how the standard error changes with the parameter value. Together they reposition the interval endpoints so coverage matches the nominal level.
How many resamples should I use?
At least 2000 resamples are recommended for BCa, since the adjusted percentiles fall in the tails of the bootstrap distribution and need a dense set of resamples to be stable.
When should I not use BCa?
With fewer than about 20 observations the jackknife acceleration estimate is unreliable, so a plain bootstrap is safer; below about 10 observations the corrections become meaningless and a permutation test is preferable. BCa also assumes independent observations, so use block or wild bootstrap for dependent data.
Sources
- Efron, B. (1987). Better Bootstrap Confidence Intervals. Journal of the American Statistical Association, 82(397), 171-185. DOI: 10.1080/01621459.1987.10478410 ↗
- DiCiccio, T. J. & Efron, B. (1996). Bootstrap Confidence Intervals. Statistical Science, 11(3), 189-228. DOI: 10.1214/ss/1032280214 ↗
How to cite this page
ScholarGate. (2026, June 1). Bias-Corrected and Accelerated Bootstrap. ScholarGate. https://scholargate.app/en/statistics/bca-bootstrap
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Bayesian BootstrapStatistics↔ compare
- Bootstrap InferenceStatistics↔ compare
- Double BootstrapStatistics↔ compare
- Permutation TestStatistics↔ compare
- Wild BootstrapStatistics↔ compare