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      "statement": "# Efron and Stein: jackknife variance through interaction decomposition\n\nEfron and Stein used orthogonal interaction decomposition to characterize the expected upward bias of the natural deletion-based jackknife variance estimate and distinguish its sample-size target and nonlinear correction limits.\n\nDeleting one observation at a time reveals how strongly a statistic changes, but the resulting dispersion needs a precise target. The paper separates the natural variance estimate for a statistic on n−1 observations from the usual scaled estimate intended for n observations. An ANOVA-like expansion decomposes the statistic into main effects and higher-order interactions. The natural estimate is unbiased for additive components but counts quadratic interactions twice and cubic interactions three times in expectation. A more general deletion-function inequality allows independent observations with different distributions and nonsymmetric functions. The source also warns that scaling to the n-observation target is not always upward biased. For a centered squared Bernoulli sample mean at n=2, every one-observation deletion has the same value, so the jackknife reports zero despite positive target variance. This does not contradict the natural n−1 theorem.\n\n## Scientific scope\n\nThe variance arguments require independent observations and finite second moments; the natural jackknife result has its stated symmetry/iid setting and separately generalized cases. Conservativeness is in expectation, not a pointwise upper bound or a confidence interval. Changing the scaling or target sample size can remove that property. Dependent observations and bootstrap proofs deferred to an unread report are outside the verified source scope.\n\n## People and systems\n\nB. Efron; C. Stein\n\n\n\n## Original sources\n\n- [Efron and Stein: jackknife variance through interaction decomposition — original scientific article](https://pdodds.w3.uvm.edu/files/papers/others/1981/efron1981a.pdf)\n- [Efron and Stein: jackknife variance through interaction decomposition — publisher deposit](https://api.crossref.org/works/10.1214%2Faos%2F1176345462)\n",
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