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September, 1989 A General Theory for Jackknife Variance Estimation
Jun Shao, C. F. J. Wu
Ann. Statist. 17(3): 1176-1197 (September, 1989). DOI: 10.1214/aos/1176347263


The delete-1 jackknife is known to give inconsistent variance estimators for nonsmooth estimators such as the sample quantiles. This well-known deficiency can be rectified by using a more general jackknife with $d$, the number of observations deleted, depending on a smoothness measure of the point estimator. Our general theory explains why jackknife works or fails. It also shows that (i) for "sufficiently smooth" estimators, the jackknife variance estimators with bounded $d$ are consistent and asymptotically unbiased and (ii) for "nonsmooth" estimators, $d$ has to go to infinity at a rate explicitly determined by a smoothness measure to ensure consistency and asymptotic unbiasedness. Improved results are obtained for several classes of estimators. In particular, for the sample $p$-quantiles, the jackknife variance estimators with $d$ satisfying $n^{1/2}/d \rightarrow 0$ and $n - d \rightarrow \infty$ are consistent and asymptotically unbiased.


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Jun Shao. C. F. J. Wu. "A General Theory for Jackknife Variance Estimation." Ann. Statist. 17 (3) 1176 - 1197, September, 1989.


Published: September, 1989
First available in Project Euclid: 12 April 2007

zbMATH: 0684.62034
MathSciNet: MR1015145
Digital Object Identifier: 10.1214/aos/1176347263

Primary: 62G05
Secondary: 62E20, 62G99

Rights: Copyright © 1989 Institute of Mathematical Statistics


Vol.17 • No. 3 • September, 1989
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