Jackknife and block bootstrap variance estimation for periodically correlated time series
Abstract
This paper develops a jackknife methodology for periodically correlated time series. We show that, as in the stationary case, the jackknife variance provides a consistent estimator of the asymptotic variance. For the overall mean and the seasonal means we derive the asymptotic bias and variance of this estimator and show that, for the seasonal means, the data have to be demeaned seasonally, since otherwise an additional bias term appears. We then relate the jackknife to the Generalized Seasonal Block Bootstrap and to the Extension of the Moving Block Bootstrap. For block lengths dividing the sample size, the bootstrap variance of the seasonal mean in the circular versions of both bootstrap schemes coincides exactly with the circular jackknife variance for every block length. The same holds for the overall mean in the Extension of the Moving Block Bootstrap case, whereas the Generalized Seasonal Block Bootstrap requires the block length to be coprime with the period. As a consequence, the block length that minimizes the mean squared error of the bootstrap variance estimator is of the same order as in the stationary case. Finally, we extend the jackknife variance estimator to differentiable functionals of the empirical distribution of blocks, establish its consistency (once it is standardised by the effective weights of the periods), and illustrate the results by a small simulation study.
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BibTeX
@misc{bertail2026jackknife,
title = {{Jackknife and block bootstrap variance estimation for periodically correlated time series}},
author = {Patrice Bertail and Anna E. Dudek},
howpublished = {HAL},
year = {2026},
month = nov,
url = {https://amu.hal.science/hal-05854527v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Jackknife and block bootstrap variance estimation for periodically correlated time series
AU - Patrice Bertail
AU - Anna E. Dudek
PY - 2026
DA - 2026/11/06
PB - HAL
UR - https://amu.hal.science/hal-05854527v1
AB - This paper develops a jackknife methodology for periodically correlated time series. We show that, as in the stationary case, the jackknife variance provides a consistent estimator of the asymptotic variance. For the overall mean and the seasonal means we derive the asymptotic bias and variance of this estimator and show that, for the seasonal means, the data have to be demeaned seasonally, since otherwise an additional bias term appears. We then relate the jackknife to the Generalized Seasonal Block Bootstrap and to the Extension of the Moving Block Bootstrap. For block lengths dividing the sample size, the bootstrap variance of the seasonal mean in the circular versions of both bootstrap schemes coincides exactly with the circular jackknife variance for every block length. The same holds for the overall mean in the Extension of the Moving Block Bootstrap case, whereas the Generalized Seasonal Block Bootstrap requires the block length to be coprime with the period. As a consequence, the block length that minimizes the mean squared error of the bootstrap variance estimator is of the same order as in the stationary case. Finally, we extend the jackknife variance estimator to differentiable functionals of the empirical distribution of blocks, establish its consistency (once it is standardised by the effective weights of the periods), and illustrate the results by a small simulation study.
ER -