Bootstrap resampling and dependent data

For independent identically distributed observations, the nonparametric bootstrap samples the original number of observations with replacement from their empirical distribution. Conditional on the observed sample, each original row has equal selection probability, including repeated equal-valued rows as separate observations. Recompute the full estimator for each resample. A parametric bootstrap instead draws from a fitted probability model. These are approximation procedures for sampling distributions; neither changes the actual observed sample size or creates additional independent experimental information.

Distinguish two sources of approximation. Even an infinite number of bootstrap replicates uses an estimated population distribution rather than the true one. A finite number of replicates adds Monte Carlo error to summaries of that conditional distribution. More replicates reduce the second problem while leaving the first. For smooth estimators under suitable regularity the approximation can improve with sample size, but sparse tails, boundary estimates, nonsmooth extrema and weak identification can defeat familiar bootstrap intervals. The sampling assumptions must therefore be checked rather than assumed from the method's name.

Measurements within one detector run can share alignment, temperature or source fluctuations. If different runs are independent clusters, resampling whole runs preserves the within-run pattern better than resampling individual readings. Equal-size cluster examples are straightforward; unequal sizes require deciding whether the target weights runs or readings. Paired measurements should normally travel together when their pairing is part of the sampling design. Resampling entries independently can erase exactly the covariance that dominates a fitted parameter's uncertainty.

For a stationary time series, nearby observations can be correlated even without natural independent clusters. The variance of its mean contains lagged covariances. An iid bootstrap ignores those terms. Block bootstraps resample stretches of consecutive observations to retain local dependence, while choosing block length balances preserved correlations against the number of available blocks. Consistency needs assumptions such as sufficiently weak dependence, appropriate moments and a suitable growing block length; no one block size or theorem covers arbitrary nonstationary or long-memory records.

A fit based on estimated calibration, selection or tuning can inherit uncertainty from those operations. A bootstrap that treats them as fixed answers a conditional question. Repeating them within each appropriate resampling scheme addresses a broader procedure, but only if the relevant data and dependence structure are represented. A fixed unknown calibration offset absent from the data cannot be learned by an ordinary row bootstrap. External calibration must enter separately or be resampled at its genuinely independent experimental level.

Report the resampling unit, scheme, number of replicates, statistic and interval rule, together with any block-length or model sensitivity. A percentile interval is not automatically a confidence interval with its nominal coverage. Exact enumeration of a tiny bootstrap distribution is valuable for understanding mechanics, but two observations or two clusters cannot establish an asymptotic coverage guarantee. Independent validation and physics-based uncertainty budgets remain necessary even when a bootstrap histogram appears smooth and numerically stable.