Hostname: page-component-6bb9c88b65-6scc2 Total loading time: 0 Render date: 2025-07-26T09:47:30.816Z Has data issue: false hasContentIssue false
  • English
  • Français

HIGHER-ORDER APPROXIMATION FOR UNCERTAINTY QUANTIFICATION IN TIME-SERIES ANALYSIS

Published online by Cambridge University Press:  22 July 2025

Annika Betken  and
Marie-Christine Düker
Annika Betken*
Affiliation:
https://ror.org/006hf6230 University of Twente
Marie-Christine Düker
Affiliation:
https://ror.org/00f7hpc57 FAU Erlangen-Nürnberg
*
Address correspondence to Annika Betken, Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, Enschede, The Netherlands, e-mail: a.betken@utwente.nl.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For time series with high temporal correlation, the empirical process converges rather slowly to its limiting distribution. Many statistics in change-point analysis, goodness-of-fit testing, and uncertainty quantification admit a representation as functionals of the empirical process and therefore inherit its slow convergence. As a result, inference based on the asymptotic distribution of those quantities is significantly affected by relatively small sample sizes. We assess the quality of higher-order approximations (HOAs) of the empirical process by deriving the asymptotic distribution of the corresponding error terms. Based on the limiting distribution of the higher-order terms, we propose a novel approach to calculate confidence intervals for statistical quantities such as the median. In a simulation study, we compare coverage rates and lengths of these confidence intervals with those based on the asymptotic distribution of the empirical process and highlight some benefits of HOAs of the empirical process.

Information

Type
ARTICLES
Information
Econometric Theory , First View , pp. 1 - 50
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Footnotes

We thank the editor, co-editor, and referees for their careful reading of the manuscript and their thoughtful comments, which led to a significant improvement of the article. Annika Betken gratefully acknowledges financial support from the Dutch Research Council (NWO) through VENI grant 212.164. Marie-Christine Düker gratefully acknowledges financial support from the National Science Foundation under grants 1934985, 1940124, 1940276, and 2114143. This research was conducted with support from the Cornell University Center for Advanced Computing, which receives funding from Cornell University, the National Science Foundation, and members of its Partner Program.

References

REFERENCES

Bagchi, P., Banerjee, M., & Stoev, S. (2016). Inference for monotone functions under short- and long-range dependence: Confidence intervals and new universal limits. Journal of the American Statistical Association , 111(516), 1634–1647.10.1080/01621459.2015.1100622CrossRefGoogle Scholar
Bai, S., & Taqqu, M. S. (2013). Multivariate limit theorems in the context of long-range dependence. Journal of Time Series Analysis , 34(6), 717–743.10.1111/jtsa.12046CrossRefGoogle Scholar
Beran, J. (1992). Statistical methods for data with long-range dependence. Statistical Science , 7(4), 404–416.Google Scholar
Beran, J., Feng, Y., Ghosh, S., & Kulik, R. (2016). Long-Memory Processes . Springer.Google Scholar
Betken, A. (2016). Testing for change-points in long-range dependent time series by means of a self-normalized Wilcoxon test. Journal of Time Series Analysis , 37(6), 785–809.Google Scholar
Betken, A. (2017). Change point estimation based on Wilcoxon tests in the presence of long-range dependence. Electronic Journal of Statistics , 11(2), 3633–3672.10.1214/17-EJS1323CrossRefGoogle Scholar
Betken, A., Giraudo, D., & Kulik, R. (2023). Change-point tests for the tail parameter of long memory stochastic volatility time series. Statistica Sinica , 33(3), 2017–2039.Google Scholar
Billingsley, P. (1968). Convergence of probability measures , Wiley Series in Probability and Mathematical Statistics: Tracts on Probability and Statistics. Wiley.Google Scholar
Bühlmann, P. (2002). Bootstraps for time series. Statistical Science , 17(1), 52–72.10.1214/ss/1023798998CrossRefGoogle Scholar
Dehling, H., Rooch, A., & Taqqu, M. (2013). Non-parametric change-point tests for long-range dependent data. Scandinavian Journal of Statistics , 40(1), 153–173.10.1111/j.1467-9469.2012.00799.xCrossRefGoogle Scholar
Dehling, H., & Taqqu, M. (1989). The empirical process of some long-range dependent sequences with an application to $U$ -Statistics. The Annals of Statistics , 17(4), 1767–1783.10.1214/aos/1176347394CrossRefGoogle Scholar
Donsker, M. (1952). Justification and extension of Doob’s heuristic approach to the Kolmogorov-Smirnov theorems. The Annals of Mathematical Statistics , 2, 277–281.10.1214/aoms/1177729445CrossRefGoogle Scholar
Doukhan, P., & Surgailis, D. (1998). Functional central limit theorem for the empirical process of short memory linear processes. Comptes Rendus de l’Académie des Sciences-Series I-Mathematics , 326(1), 87–92.Google Scholar
Dudley, R. (1978). Central limit theorems for empirical measures. The Annals of Probability , 6(6), 899–929.10.1214/aop/1176995384CrossRefGoogle Scholar
El Ktaibi, F., & Ivanoff, B. G. (2016). Change-point detection in the marginal distribution of a linear process. Electronic Journal of Statistics , 10(2), 3945–3985.10.1214/16-EJS1215CrossRefGoogle Scholar
Fang, Y., Xu, J., & Yang, L. (2018). Online bootstrap confidence intervals for the stochastic gradient descent estimator. The Journal of Machine Learning Research , 19(1), 3053–3073.Google Scholar
Hall, P., Jing, B.-Y., & Lahiri, S. (1998). On the sampling window method for long-range dependent data. Statistica Sinica , 8, 1189–1204.Google Scholar
Hoga, Y. (2019). Confidence intervals for conditional tail risk measures in ARMA–GARCH models. Journal of Business & Economic Statistics , 37(4), 613–624.10.1080/07350015.2017.1401543CrossRefGoogle Scholar
Hössjer, O., & Mielniczuk, J. (1995). Delta method for long-range dependent observations. Journal of Nonparametric Statistics , 5(1), 75–82.10.1080/10485259508832635CrossRefGoogle Scholar
Huang, Y., & Shao, X. (2016). Coverage bound for fixed- $b$ subsampling and generalized subsampling for time series. Statistica Sinica , 26(4), 1499–1524.Google Scholar
Ivanoff, B. G. (1980). The function space $D([0,\infty)^{q},E)$ . Canadian Journal of Statistics , 8(2), 179–191.10.2307/3315230CrossRefGoogle Scholar
Kiefer, J. (1972). Skorohod embedding of multivariate RV’s, and the sample DF. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete , 24(1), 1–35.10.1007/BF00532460CrossRefGoogle Scholar
Kim, S., Zhao, Z., & Shao, X. (2015). Nonparametric functional central limit theorem for time series regression with application to self-normalized confidence interval. Journal of Multivariate Analysis , 133, 277–290.10.1016/j.jmva.2014.09.017CrossRefGoogle ScholarPubMed
Koul, H., & Surgailis, D. (2002). Asymptotic expansion of the empirical process of long memory moving averages. In H. Dehling, T. Mikosch, & M. Sørensen (Eds.), Empirical process techniques for dependent data (213–239). Springer.10.1007/978-1-4612-0099-4_7CrossRefGoogle Scholar
Lahiri, S. N. (1993). On the moving block bootstrap under long range dependence. Statistics & Probability Letters , 18(5), 405–413.CrossRefGoogle Scholar
Massah, M., & Kantz, H. (2016). Confidence intervals for time averages in the presence of long-range correlations, a case study on earth surface temperature anomalies. Geophysical Research Letters , 43(17), 9243–9249.10.1002/2016GL069555CrossRefGoogle Scholar
Müller, D. W. (1970). On Glivenko-Cantelli convergence. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete , 16(3), 195–210.10.1007/BF00534596CrossRefGoogle Scholar
Nordman, D., Bunzel, H., & Lahiri, S. (2013). A nonstandard empirical likelihood for time series. The Annals of Statistics , 41(6), 3050–3073.10.1214/13-AOS1174CrossRefGoogle Scholar
Nordman, D., Sibbertsen, P., & Lahiri, S. N. (2007). Empirical likelihood confidence intervals for the mean of a long-range dependent process. Journal of Time Series Analysis , 28(4), 576–599.10.1111/j.1467-9892.2006.00526.xCrossRefGoogle Scholar
Nourdin, I., & Rosiński, J. (2014). Asymptotic independence of multiple Wiener–Itô integrals and the resulting limit laws. The Annals of Probability , 42(2), 497–526.10.1214/12-AOP826CrossRefGoogle Scholar
Pipiras, V., & Taqqu, M. (2017). Long-Range Dependence and Self-Similarity , Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press.Google Scholar
Purwanto, A., & Sudargini, Y. (2021). Partial least squares structural squation modeling (pls-sem) analysis for social and management research: A literature review. Journal of Industrial Engineering & Management Research , 2(4), 114–123.Google Scholar
Rust, H. W., Kallache, M., Schellnhuber, H. J., & Kropp, J. P. (2010). Confidence intervals for flood return level estimates assuming long-range dependence. In J. Kropp & H.-J. Schellnhuber (Eds.), Extremis: Disruptive events and trends in climate and hydrology (60–88). Springer.Google Scholar
Shao, X. (2010). A self-normalized approach to confidence interval construction in time series. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 72(3), 343–366.10.1111/j.1467-9868.2009.00737.xCrossRefGoogle Scholar
Shorack, G., & Wellner, J. (2009). Empirical processes with applications to statistics . SIAM.10.1137/1.9780898719017CrossRefGoogle Scholar
Taqqu, M. S. (1977). Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete , 40(3), 203–238.10.1007/BF00736047CrossRefGoogle Scholar
Taqqu, M. S., Teverovsky, V., & Willinger, W. (1995). Estimators for long-range dependence: An empirical study. Fractals , 3(4), 785–798.CrossRefGoogle Scholar
Tewes, J. (2018). Block bootstrap for the empirical process of long-range dependent data. Journal of Time Series Analysis , 39(1), 28–53.10.1111/jtsa.12256CrossRefGoogle Scholar
Van der Vaart, A. (2000). Asymptotic statistics (vol. 3). Cambridge University Press.Google Scholar
Wellner, J. & van der Vaart, A. (2013). Weak convergence and empirical processes: with applications to statistics . Springer Science & Business Media.Google Scholar
Weron, R. (2002). Estimating long-range dependence: finite sample properties and confidence intervals. Physica A: Statistical Mechanics and its Applications , 312(1), 285–299.10.1016/S0378-4371(02)00961-5CrossRefGoogle Scholar
Youndjé, E., & Vieu, P. (2006). A note on quantile estimation for long-range dependent stochastic processes. Statistics & Probability Letters , 76(2), 109–116.10.1016/j.spl.2005.06.015CrossRefGoogle Scholar