Abstract
Frequency domain methods form a ubiquitous part of the statistical toolbox for time-series analysis. In recent years, considerable interest has been given to the development of new spectral methodology and tools capturing dynamics in the entire joint distributions, and thus avoiding the limitations of classical, -based spectral methods. Most of the spectral concepts proposed in that literature suffer from one major drawback, though: their estimation requires the choice of a smoothing parameter, which has a considerable impact on estimation quality and poses challenges for statistical inference. In this paper, associated with the concept of a copula-based spectrum, we introduce the notion of a copula spectral distribution function or integrated copula spectrum. This integrated copula spectrum retains the advantages of copula-based spectra but can be estimated without the need for smoothing parameters. We provide such estimators, along with a thorough theoretical analysis, based on a functional central limit theorem, of their asymptotic properties. We leverage these results to test various hypotheses that cannot be addressed by classical spectral methods, such as the lack of time reversibility or asymmetry in tail dynamics.
Funding Statement
This work has been supported in part by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823, Teilprojekt A1,C1) of the German Research Foundation (DFG). Yuichi Goto was supported by JSPS Grant-in-Aid for the Research Activity Start-up under Grant Number JP21K20338. Stanislav Volgushev was partially supported by a discovery grant from NSERC of Canada.
Acknowledgments
The authors would like to thank the Associate Editor and a referee for their careful reading and instructive comments, which helped to improve the quality of our manuscript.
The first three authors contributed equally to the paper and are listed alphabetically.
The authors are grateful to Brendan K. Beare and Juwon Seo for kindly sharing their Matlab codes for Section 5.2.
Citation
Yuichi Goto. Tobias Kley. Ria Van Hecke. Stanislav Volgushev. Holger Dette. Marc Hallin. "The integrated copula spectrum." Ann. Statist. 50 (6) 3563 - 3591, December 2022. https://doi.org/10.1214/22-AOS2240
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