Publication record · 18.cifr/1979.efron.bootstrap

Bootstrap Methods: Another Look at the Jackknife

v1.0.0

B. Efron (Stanford University)

RAI18.cifr/1979.efron.bootstrap

The Annals of Statistics· 1979· doi:10.1214/aos/1176344552

We discuss the following problem: given a random sample X = (X_1, X_2, ..., X_n) from an unknown probability distribution F, estimate the sampling distribution of some prespecified random variable R(X, F), on the basis of the observed data x. (Standard jackknife theory gives an approximate mean and variance in the case R(X,F) = theta(F_hat) - theta(F).) Our main tool is to resampling x in a simple way. The resulting 'bootstrap' distribution is shown to be consistent for a variety of applications.

bootstrapjackkniferesamplingstandard errorconfidence intervalsnonparametric

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Before the bootstrap, estimating the standard error of a complex statistic required case-by-case analytic derivations or Taylor-series linearization that breaks down for nonsmooth statistics. The jackknife partially addressed this but requires differentiability. The bootstrap provides a single computational procedure that works for any statistic regardless of its analytical tractability.

Novelty

Efron introduced the bootstrap as a general-purpose resampling method that estimates the sampling distribution of any statistic by drawing B random samples with replacement from the original data. Unlike the jackknife, which uses leave-one-out subsamples and requires linearizability, the bootstrap makes no assumptions about the functional form of the statistic. The paper showed the bootstrap reduces to the jackknife asymptotically, unifying both under a single framework.

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