The Jackknife and the Bootstrap for General Stationary Observations

• Published in: The Annals of statistics 1989-09, Vol.17 (3), p.1217-1241
• Main Author: Kunsch, Hans R.
• Format: Article
• Language: English
• Subjects: 62G05; 62G15; 62M10; Approximation; Arithmetic mean; bootstrap; Covariance; Estimation methods; Estimators; Exact sciences and technology; influence function; jackknife; Mathematical theorems; Mathematics; Nonparametric inference; Parametric models; Probability and statistics; Sample variance; Sciences and techniques of general use; Standard error; Statistical variance; Statistics; statistics defined by functionals; time series; Variance estimation
• ISSN: 0090-5364; 2168-8966
• DOI: 10.1214/aos/1176347265

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) → ∞ and l(n)/n → 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.