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Bootstrapping (statistics)
In statistics, bootstrapping can refer to any test or metric that relies on random sampling with replacement. Bootstrapping allows assigning measures of accuracy (defined in terms of bias, variance, confidence intervals, prediction error or some other such measure) to sample estimates.〔 (software )〕〔Second Thoughts on the Bootstrap - Bradley Efron, 2003〕 This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods.〔〔Weisstein, Eric W. "Bootstrap Methods." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BootstrapMethods.html〕 Generally, it falls in the broader class of resampling methods.
Bootstrapping is the practice of estimating properties of an estimator (such as its variance) by measuring those properties when sampling from an approximating distribution. One standard choice for an approximating distribution is the empirical distribution function of the observed data. In the case where a set of observations can be assumed to be from an independent and identically distributed population, this can be implemented by constructing a number of resamples with replacement, of the observed dataset (and of equal size to the observed dataset).
It may also be used for constructing hypothesis tests. It is often used as an alternative to statistical inference based on the assumption of a parametric model when that assumption is in doubt, or where parametric inference is impossible or requires complicated formulas for the calculation of standard errors.
==History==
The bootstrap was published by Bradley Efron in "Bootstrap methods: another look at the jackknife" (1979).〔(Notes for Earliest Known Uses of Some of the Words of Mathematics: Bootstrap ) (John Aldrich)〕〔(Earliest Known Uses of Some of the Words of Mathematics (B) ) (Jeff Miller)〕 It was inspired by earlier work on the jackknife.〔Quenouille M (1949) Approximate tests of correlation in time-series. J Roy Statist Soc Ser B 11 68–84〕〔Tukey J (1958) Bias and confidence in not-quite large samples (abstract). Ann Math Statist 29 614〕〔Jaeckel L (1972) The infinitesimal jackknife. Memorandum MM72-1215-11, Bell Lab〕 Improved estimates of the variance were developed later.〔Bickel P, Freeman D (1981) Some asymptotic theory for the bootstrap. Ann Statist 9 1196–1217〕〔Singh K (1981) On the asymptotic accuracy of Efron’s bootstrap. Ann Statist 9 1187–1195〕 A Bayesian extension was developed in 1981.〔Rubin D (1981). The Bayesian bootstrap. Ann Statist 9 130–134〕 The bias-corrected and accelerated (BCa) bootstrap was developed by Efron in 1987,〔 and the ABC procedure in 1992.〔Diciccio T, Efron B (1992) More accurate confidence intervals in exponential families. Biometrika 79 231–245〕
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