Bootstrap CI is preferable when: (1) the statistic has no simple closed-form standard error (e.g., the median, IQR, or a regression R²); (2) the population distribution is clearly non-normal or heavy-tailed; (3) the sample size is small, making asymptotic normality suspect; or (4) you want a "sanity check" on parametric results. The standard t-interval for the mean is reliable for n ≥ 30 under moderate departures from normality, but bootstrap is always valid under weaker assumptions.

The percentile bootstrap can be biased or have incorrect coverage if the sampling distribution of θ̂ is skewed or if θ̂ is a biased estimator of θ. The bias-corrected and accelerated (BCa) method, also by Efron, adjusts the quantile levels used to extract the CI endpoints using a bias correction z₀ (based on how often θ̂* < θ̂) and an acceleration constant a (based on the jackknife influence values). BCa CIs have better theoretical coverage and are recommended for skewed statistics.

Standard i.i.d. bootstrap fails for time series because it destroys temporal correlation. Instead, block bootstrap methods resample consecutive blocks of observations (e.g., blocks of 20 data points) to preserve autocorrelation structure. Common variants include the moving block bootstrap, circular block bootstrap, and stationary bootstrap (random block lengths from a geometric distribution). For spatial data, spatial bootstrap or subsampling procedures are needed.

The jackknife (Quenouille, 1949; Tukey, 1958) estimates bias and variance by sequentially leaving out one observation at a time, computing the statistic on the n−1 remaining values, and averaging or differencing the results. It is computationally cheaper than bootstrap (only n resamples vs B) but is limited to smooth statistics: it fails for the median and quantiles because small perturbations do not change the rank-order statistic. The bootstrap subsumes the jackknife as a special case.

Bootstrap CI width scales approximately as 1/√n for statistics with finite variance (like the mean). Doubling sample size from 30 to 120 halves the CI width. For heavy-tailed distributions (Cauchy has no finite variance), this 1/√n scaling breaks down and bootstrap CIs can be inconsistent. The simulator lets you observe this directly: try n = 10 vs n = 100 with the Cauchy-like ("heavy") distribution and compare how the CI width changes.

Yes — this is one of bootstrap's most popular modern applications. To estimate the variance of cross-validation accuracy, the .632 bootstrap evaluates each bootstrap model on the out-of-bag observations (the ~37% not drawn), combining with the training accuracy: erroṙ = 0.368 × training_error + 0.632 × out-of-bag_error. The .632+ method further adjusts for overfitting. Scikit-learn and R's caret package offer bootstrap-based model evaluation.

Bootstrap is inconsistent for statistics that depend on extreme order statistics (e.g., the sample maximum or minimum), because the maximum of a resample can never exceed the maximum of the original sample. It also requires that the original sample is representative of the population — if the sample has selection bias, the bootstrap CI inherits that bias. Finally, computational cost scales with B and n, which can be prohibitive for slow model fits or very large datasets without sub-sampling.

The plug-in principle says: estimate any population quantity by substituting the empirical distribution function F̂(x) — the step function that places mass 1/n at each observed value — for the true distribution F(x). Bootstrap sampling from the original data is equivalent to sampling from F̂. Any population quantity, whether a mean, a quantile, or a complex functional, can then be estimated by computing the same quantity under F̂, with uncertainty characterised by the variability across bootstrap resamples.