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Estimation of the variance of partial sums of dependent processes

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  • Dehling, Herold
  • Fried, Roland
  • Sharipov, Olimjon Sh.
  • Vogel, Daniel
  • Wornowizki, Max

Abstract

We study subsampling estimators for the limit variance σ2=V ar(X1)+2∑k=2∞Cov(X1,Xk) of partial sums of a stationary stochastic process (Xk)k≥1. We establish L2-consistency of a non-overlapping block resampling method. Our results apply to processes that can be represented as functionals of strongly mixing processes. Motivated by recent applications to rank tests, we also study estimators for the series V ar(F(X1))+2∑k=2∞Cov(F(X1),F(Xk)), where F is the distribution function of X1. Simulations illustrate the usefulness of the proposed estimators and of a mean squared error optimal rule for the choice of the block length.

Suggested Citation

  • Dehling, Herold & Fried, Roland & Sharipov, Olimjon Sh. & Vogel, Daniel & Wornowizki, Max, 2013. "Estimation of the variance of partial sums of dependent processes," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 141-147.
    • Handle: RePEc:eee:stapro:v:83:y:2013:i:1:p:141-147
    DOI: 10.1016/j.spl.2012.08.012
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    References listed on IDEAS

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    1. Peligard, Magda & Suresh, Ram, 1995. "Estimation of variance of partial sums of an associated sequence of random variables," Stochastic Processes and their Applications, Elsevier, vol. 56(2), pages 307-319, April.
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    4. Robert M. De Jong & James Davidson, 2000. "Consistency of Kernel Estimators of Heteroscedastic and Autocorrelated Covariance Matrices," Econometrica, Econometric Society, vol. 68(2), pages 407-424, March.
    5. Dehling, Herold & Fried, Roland, 2012. "Asymptotic distribution of two-sample empirical U-quantiles with applications to robust tests for shifts in location," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 124-140.
    6. Peligrad, M. & Shao, Q. M., 1995. "Estimation of the Variance of Partial Sums for [rho]-Mixing Random Variables," Journal of Multivariate Analysis, Elsevier, vol. 52(1), pages 140-157, January.
    7. Shao, Qi-Man & Yu, Hao, 1993. "Bootstrapping the sample means for stationary mixing sequences," Stochastic Processes and their Applications, Elsevier, vol. 48(1), pages 175-190, October.
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    Cited by:

    1. Sayar Karmakar & Marek Chudý & Wei Biao Wu, 2022. "Long‐term prediction intervals with many covariates," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(4), pages 587-609, July.
    2. Sayar Karmakar & Marek Chudy & Wei Biao Wu, 2020. "Long-term prediction intervals with many covariates," Papers 2012.08223, arXiv.org, revised Sep 2021.
    3. Carina Gerstenberger, 2021. "Robust discrimination between long‐range dependence and a change in mean," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(1), pages 34-62, January.
    4. M. Chudý & S. Karmakar & W. B. Wu, 2020. "Long-term prediction intervals of economic time series," Empirical Economics, Springer, vol. 58(1), pages 191-222, January.

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