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Nonparametric Bootstrap Resampling (Efron, 1979)

The nonparametric bootstrap is a resampling procedure introduced by Efron (1979) for estimating the sampling distribution of a statistic without parametric assumptions about the underlying population. Given an observed sample X=(x1,,xn)X = (x_1,\dots,x_n) and a statistic θ^=T(X)\hat\theta = T(X), the bootstrap repeatedly draws resamples X(b)X^{*(b)} of size nn with replacement from XX and recomputes θ^(b)=T(X(b))\hat\theta^{*(b)} = T(X^{*(b)}) for b=1,,Bb = 1,\dots,B. The empirical distribution of {θ^(b)}\{\hat\theta^{*(b)}\} approximates the sampling distribution of θ^\hat\theta and yields standard errors, percentile confidence intervals, and bias estimates. Treating the empirical distribution of the observed sample as a plug-in estimate of the population is the central idea that underlies all later bootstrap variants, including paired versions for two-sample comparisons.

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Updated 2026-05-16

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