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A general theory for jackknife variance estimation. (English) Zbl 0684.62034
Summary: 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 \(n^{1/2}/d\to 0\) and n-d\(\to \infty\) are consistent and asymptotically unbiased.

MSC:
62G05 Nonparametric estimation
62G99 Nonparametric inference
62E20 Asymptotic distribution theory in statistics
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