= Jackknife resampling
{wiki}
The delete-one form of <jackknife resampling> recalculates a <statistic> after deleting each observation in turn. It uses one observed <statistical sample> to investigate how sensitive an <estimator> is to its individual observations, and supplies both a <jackknife variance estimator> and <jackknife bias correction>. The calculations are deterministic once the data are fixed: no random draws are needed.
In the basic setting, $X_1,\ldots,X_n$ are <independent and identically distributed random variables>, with <sample size> $n>1$. Let $x_1,\ldots,x_n$ be the observed values and $\widehat\theta_n=t_n(x_1,\ldots,x_n)$ a scalar <estimator> of an <estimand> $\theta$. Apply the same estimation rule to each reduced <statistical sample>, using its new <sample size> $n-1$ wherever the rule requires it:
$$
\widehat\theta_{(-i)}=t_{n-1}(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n),\qquad
i=1,\ldots,n,\qquad
\overline\theta_J=\frac1n\sum_{i=1}^n\widehat\theta_{(-i)}.
$$
Every index is deleted exactly once; an unusual value is treated by the same rule as every other value. The <mean> $\overline\theta_J$ summarizes the deletion estimates. They overlap in $n-2$ observations, so they are generally not <independent random variables> representing fresh repetitions of the original experiment. Cosma Shalizi's https://stat.cmu.edu/~cshalizi/490/23/2023-11-29-jackknife.pdf[Carnegie Mellon jackknife notes, Sections 1–3 and 5]{ref} explain this construction and its scaling.
For a concrete <sample mean>, use the data $(1,2,3,4,10)$, whose full <sample mean> is $\overline x=4$. The five deletion <sample means>, in observation order, are
$$
(\widehat\theta_{(-1)},\ldots,\widehat\theta_{(-5)})
=(4.75,4.5,4.25,4,2.5),\qquad \overline\theta_J=4.
$$
Deleting $1$ raises the <sample mean> by $0.75$, while deleting $10$ lowers the <sample mean> by $1.5$. Thus $10$ has the largest deletion effect in this <statistical sample>. This is a descriptive sensitivity calculation, not a rule for removing that observation: the full <sample mean> remains $4$.
\Image[/jackknife-leave-one-out.png]
{title=Delete each observation once and recompute the <sample mean>}
{description=The left panel shows all five reduced <statistical samples> from $(1,2,3,4,10)$; each crossed entry is omitted only for that row. The right panel shows their <sample means> against the full <sample mean> $4$. The orange row makes the deletion effect of $10$ visible.}
Early literature includes M. H. Quenouille's https://academic.oup.com/jrsssb/article/11/1/68/7026551[Approximate Tests of Correlation in Time-Series, JRSS B 11(1), 68–84 (1949)]{ref} and John W. Tukey's one-paragraph abstract https://www.jstor.org/stable/2237363[Bias and Confidence in Not-Quite Large Samples, Annals of Mathematical Statistics 29(2), 614 (1958)]{ref}, within the issue's collected abstracts. Shalizi's https://stat.cmu.edu/~cshalizi/490/23/2023-11-29-jackknife.pdf[historical notes and bibliography]{ref} discuss the later development of <jackknife bias correction> and the <jackknife variance estimator>.
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