The Annals of Statistics
- Ann. Statist.
- Volume 17, Number 3 (1989), 1217-1241.
The Jackknife and the Bootstrap for General Stationary Observations
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.
Ann. Statist. Volume 17, Number 3 (1989), 1217-1241.
First available in Project Euclid: 12 April 2007
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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.