The Jackknife and the Bootstrap for General Stationary Observations (original) (raw)

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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

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 lll 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)rightarrowinftyl = l(n) \rightarrow \inftyl=l(n)rightarrowinfty and l(n)/nrightarrow0l(n)/n \rightarrow 0l(n)/nrightarrow0. 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 lll 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. https://doi.org/10.1214/aos/1176347265

Information

Published: September, 1989

First available in Project Euclid: 12 April 2007

Digital Object Identifier: 10.1214/aos/1176347265

Subjects:

Primary: 62G05

Secondary: 62G15, 62M10

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