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.
Publié le : 1989-09-14
Classification:
Variance estimation,
jackknife,
bootstrap,
statistics defined by functionals,
time series,
influence function,
62G05,
62G15,
62M10
@article{1176347265,
author = {Kunsch, Hans R.},
title = {The Jackknife and the Bootstrap for General Stationary Observations},
journal = {Ann. Statist.},
volume = {17},
number = {1},
year = {1989},
pages = { 1217-1241},
language = {en},
url = {http://dml.mathdoc.fr/item/1176347265}
}
Kunsch, Hans R. The Jackknife and the Bootstrap for General Stationary Observations. Ann. Statist., Tome 17 (1989) no. 1, pp. 1217-1241. http://gdmltest.u-ga.fr/item/1176347265/