April 2024 A general framework to quantify deviations from structural assumptions in the analysis of nonstationary function-valued processes
Anne van Delft, Holger Dette
Author Affiliations +
Ann. Statist. 52(2): 550-579 (April 2024). DOI: 10.1214/24-AOS2358


We present a general theory to quantify the uncertainty from imposing structural assumptions on the second-order structure of nonstationary Hilbert space-valued processes, which can be measured via functionals of time-dependent spectral density operators. The second-order dynamics are well known to be elements of the space of trace class operators, the latter is a Banach space of type 1 and of cotype 2, which makes the development of statistical inference tools more challenging. A part of our contribution is to obtain a weak invariance principle as well as concentration inequalities for (functionals of) the sequential time-varying spectral density operator. In addition, we introduce deviation measures in the nonstationary context, and derive corresponding estimators that are asymptotically pivotal. We then apply this framework and propose statistical methodology to investigate the validity of structural assumptions for nonstationary functional data, such as low-rank assumptions in the context of time-varying dynamic fPCA and principle separable component analysis, deviations from stationarity with respect to the square root distance, and deviations from zero functional canonical coherency.

Funding Statement

Anne van Delft was partially supported by NSF grant DMS 23-11338.
This work was partially supported by the DFG Research unit 5381 Mathematical Statistics in the Information Age, project number 460867398.


The authors would like to thank the associate editor and two anonymous referees for their constructive comments on an earlier version of this paper.


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Anne van Delft. Holger Dette. "A general framework to quantify deviations from structural assumptions in the analysis of nonstationary function-valued processes." Ann. Statist. 52 (2) 550 - 579, April 2024. https://doi.org/10.1214/24-AOS2358


Received: 1 September 2023; Published: April 2024
First available in Project Euclid: 9 May 2024

Digital Object Identifier: 10.1214/24-AOS2358

Primary: 60B12 , 62M10
Secondary: 60F17 , 60J05 , 62M15

Keywords: Concentration inequalities , Functional data analysis , Locally stationary processes , Martingale theory , nuclear operators , principle components , relevant hypotheses

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 2 • April 2024
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