## The jackknife and the bootstrap for general stationary observations.(English)Zbl 0684.62035

Summary: 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 $$\ell$$ 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 $$\ell =\ell (n)\to \infty$$ and $$\ell (n)/n\to 0.$$
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 $$\ell$$ randomly with replacement among the blocks of observations. The procedures are illustrated by using the sunspot numbers and some simulated data.

### MSC:

 62G05 Nonparametric estimation 62G15 Nonparametric tolerance and confidence regions 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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