What is recorded here?
Efron 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. Deleting 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.
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Scientific scope
The 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.
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- Efron and Stein: jackknife variance through interaction decomposition — original scientific article
- Efron and Stein: jackknife variance through interaction decomposition — publisher deposit
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People and systems
B. Efron; C. Stein
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English catalogue narrative
Efron and Stein: jackknife variance through interaction decomposition
Efron 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.
Deleting 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.
Scientific scope
The 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.
People and systems
B. Efron; C. Stein