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Both bootstrap and jackknife methods can be used to estimate bias and standard error of an estimate and mechanisms of both resampling methods are not huge different: sampling with replacement vs. leave out one observation at a time. However, jackknife is not as popular as bootstrap in research and practice.

Is there any obvious advantage of using bootstrap instead of using jackknife?

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    $\begingroup$ Just as a matter of history, I learned about the jackknife in the early 1970s, when statistics was still largely done on a yellow pad. (Computer time was too expensive!) If memory serves, it was promoted by John Tukey. $\endgroup$ Mar 3, 2017 at 11:00

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Bootstrapping is a superior technique and can be used pretty much anywhere jackknifing has been used. Jackknifing is much older (perhaps ~20 years); it's main advantage in the days when computing power was limited, was that it's computationally much simpler. However, the bootstrap provides information about the whole sampling distribution, and can offer greater precision. The jackknife is still useful in outlier detection, for example in calculating dfbeta (the change in a parameter estimate when a data point is dropped).

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    $\begingroup$ But perhaps also see @Benjamin's answer here (stats.stackexchange.com/questions/96739/…) as a case where a jackknife is still useful. Jackknifes are also still used (it seems) in estimating $a$ when calculating BCa confidence intervals. $\endgroup$ May 7, 2014 at 14:43
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    $\begingroup$ @gung could you provide more details or references for your claims that the bootstrap provides information about the whole sampling distribution (the jackknife does not?) and that it is more precise? $\endgroup$
    – mloning
    Oct 2, 2018 at 11:54

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