## The Annals of Statistics

### A General Theory for Jackknife Variance Estimation

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 17, Number 3 (1989), 1176-1197.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176347263

Digital Object Identifier
doi:10.1214/aos/1176347263

Mathematical Reviews number (MathSciNet)
MR1015145

Zentralblatt MATH identifier
0684.62034

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62E20: Asymptotic distribution theory 62G99: None of the above, but in this section

#### Citation

Shao, Jun; Wu, C. F. J. A General Theory for Jackknife Variance Estimation. Ann. Statist. 17 (1989), no. 3, 1176--1197. doi:10.1214/aos/1176347263. https://projecteuclid.org/euclid.aos/1176347263