The Annals of Statistics

The Jackknife and the Bootstrap for General Stationary Observations

Hans R. Kunsch

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Abstract

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.

Article information

Source
Ann. Statist. Volume 17, Number 3 (1989), 1217-1241.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176347265

Digital Object Identifier
doi:10.1214/aos/1176347265

Mathematical Reviews number (MathSciNet)
MR1015147

Zentralblatt MATH identifier
0684.62035

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62G15: Tolerance and confidence regions 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Keywords
Variance estimation jackknife bootstrap statistics defined by functionals time series influence function

Citation

Kunsch, Hans R. The Jackknife and the Bootstrap for General Stationary Observations. Ann. Statist. 17 (1989), no. 3, 1217--1241. doi:10.1214/aos/1176347265. https://projecteuclid.org/euclid.aos/1176347265.


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