November 2022 Pivotal tests for relevant differences in the second order dynamics of functional time series
Anne van Delft, Holger Dette
Author Affiliations +
Bernoulli 28(4): 2260-2293 (November 2022). DOI: 10.3150/21-BEJ1418


Motivated by the need to statistically quantify differences between modern (complex) data-sets which commonly result as high-resolution measurements of stochastic processes varying over a continuum, we propose novel testing procedures to detect relevant differences between the second order dynamics of two functional time series. In order to take the between-function dynamics into account that characterize this type of functional data, a frequency domain approach is taken. Test statistics are developed to compare differences in the spectral density operators and in the primary modes of variation as encoded in the associated eigenelements. Under mild moment conditions, we show convergence of the underlying statistics to Brownian motions and construct pivotal test statistics. The latter is essential because the nuisance parameters can be unwieldy and their robust estimation infeasible, especially if the two functional time series are dependent. In addition to these novel features, the properties of the tests are robust to any choice of frequency band enabling also to compare energy contents at a single frequency. The finite sample performance of the tests are verified through a simulation study and are illustrated with an application to fMRI data.

Funding Statement

This work has been supported by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823, Teilprojekt A1, C1) of the German Research Foundation (DFG).


The authors would like to thank the referees and the editor for their constructive comments on the first version of this manuscript.


Download Citation

Anne van Delft. Holger Dette. "Pivotal tests for relevant differences in the second order dynamics of functional time series." Bernoulli 28 (4) 2260 - 2293, November 2022.


Received: 1 October 2020; Published: November 2022
First available in Project Euclid: 17 August 2022

zbMATH: 07594059
MathSciNet: MR4474543
Digital Object Identifier: 10.3150/21-BEJ1418

Keywords: functional data , Martingale theory , relevant tests , self-normalization , spectral analysis , time series


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Vol.28 • No. 4 • November 2022
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