Open Access
September, 1989 The Jackknife and the Bootstrap for General Stationary Observations
Hans R. Kunsch
Ann. Statist. 17(3): 1217-1241 (September, 1989). DOI: 10.1214/aos/1176347265


We extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence. We do not attempt a reduction to i.i.d. values. The jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of $l$ consecutive data once. In the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero. Under appropriate conditions consistency is obtained if $l = l(n) \rightarrow \infty$ and $l(n)/n \rightarrow 0$. General statistics are approximated by an arithmetic mean. In regular cases this approximation determines the asymptotic behavior. Bootstrap replicates are constructed by selecting blocks of length $l$ randomly with replacement among the blocks of observations. The procedures are illustrated by using the sunspot numbers and some simulated data.


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Hans R. Kunsch. "The Jackknife and the Bootstrap for General Stationary Observations." Ann. Statist. 17 (3) 1217 - 1241, September, 1989.


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

zbMATH: 0684.62035
MathSciNet: MR1015147
Digital Object Identifier: 10.1214/aos/1176347265

Primary: 62G05
Secondary: 62G15 , 62M10

Keywords: bootstrap , influence function , jackknife , statistics defined by functionals , time series , variance estimation

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 3 • September, 1989
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