A nonparametric statistical method that approximates the sampling distribution of an estimator by repeatedly drawing samples with replacement from the observed dataset and recalculating the estimator on each resample.

Principle

Treat the observed sample as a proxy for the population distribution and use resampling with replacement to generate variability estimates for statistics without relying on parametric assumptions.

Demonstration

Given n observed data points, draw B bootstrap samples (each of size n with replacement), compute the sample median for each resample, and use the empirical distribution of these medians to form confidence intervals and bias estimates.

Misapplication

Using the simple bootstrap on strongly dependent time series or on very small samples and interpreting bootstrap intervals as exact when exchangeability or representativity conditions fail.

Consequence

Enables approximate inference (standard errors, confidence intervals, bias correction) for complex estimators where analytic sampling distributions are unavailable or intractable.

Reversal

Analytical parametric inference derives sampling properties from assumed distributional models and closed-form formulas rather than approximating them by resampling the observed data.

Boundary

Assumes the observed sample is representative and observations are exchangeable or that dependence is modeled (e.g., block bootstrap); excludes naive application to non-exchangeable data without adjustment.

Semantic Tension

Competes with Bayesian and asymptotic parametric methods: bootstrap provides frequentist, data-driven uncertainty quantification while Bayesian posterior summaries incorporate prior information and model structure.

Synthesis

Bootstrap resampling approximates the sampling variability of estimators by repeatedly sampling the observed data with replacement, producing empirical distributions used for uncertainty quantification without strong parametric modeling.