The Jackknife and the Bootstrap for General Stationary Observations

TLDR

General statistics are approximated by an arithmetic mean. The study extends jackknife and bootstrap standard‑error estimation to general stationary sequences. The jackknife computes the sample variance of block‑deleted replicates, while the bootstrap generates block‑resampled replicates, without reducing to i.i.d. observations.

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.