The bootstrap approximates a sampling distribution by repeatedly sampling with replacement from the empirical distribution.
A bootstrap confidence interval uses the conditional resampling distribution of an estimator to choose its endpoints. Basic intervals invert bootstrap error quantiles; percentile intervals use quantiles of the resampled estimates themselves. Their asymptotic coverage requires conditions appropriate to the statistic and interval construction.
A basic bootstrap interval reflects quantiles of the bootstrap estimation error about the observed estimate. For the sample mean it uses , where is a conditional quantile of .
For iid observations with finite positive variance, the conditional distribution of tends to the same normal limit as . Continuity of this limit gives uniform convergence of distribution functions and consistency of interior quantiles.
A bootstrap sample of size is drawn with replacement from the observed data points. Some observations occur repeatedly and about a proportion are omitted.
A parametric bootstrap simulates repeated datasets from a fitted parametric model and refits the procedure to each dataset. The resulting empirical distribution approximates the sampling distribution under that fitted model.
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Bootstrapping is a statistical resampling technique used to estimate the distribution of a sample statistic by repeatedly resampling with replacement from the data set. The central idea is to create multiple simulated samples (called "bootstrap samples"), allowing for the assessment of variability and confidence intervals of the statistic of interest without relying on strong parametric assumptions. ### Key Steps in Bootstrapping: 1. **Original Sample**: Start with an observed dataset of size \( n \).