Open Access
October 2020 Testing for stationarity of functional time series in the frequency domain
Alexander Aue, Anne van Delft
Ann. Statist. 48(5): 2505-2547 (October 2020). DOI: 10.1214/19-AOS1895


Interest in functional time series has spiked in the recent past with papers covering both methodology and applications being published at a much increased pace. This article contributes to the research in this area by proposing a new stationarity test for functional time series based on frequency domain methods. The proposed test statistics is based on joint dimension reduction via functional principal components analysis across the spectral density operators at all Fourier frequencies, explicitly allowing for frequency-dependent levels of truncation to adapt to the dynamics of the underlying functional time series. The properties of the test are derived both under the null hypothesis of stationary functional time series and under the smooth alternative of locally stationary functional time series. The methodology is theoretically justified through asymptotic results. Evidence from simulation studies and an application to annual temperature curves suggests that the test works well in finite samples.


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Alexander Aue. Anne van Delft. "Testing for stationarity of functional time series in the frequency domain." Ann. Statist. 48 (5) 2505 - 2547, October 2020.


Received: 1 October 2017; Revised: 1 April 2019; Published: October 2020
First available in Project Euclid: 19 September 2020

MathSciNet: MR4152111
Digital Object Identifier: 10.1214/19-AOS1895

Primary: 62G99 , 62H99
Secondary: 62M10 , 62M15 , 91B84

Keywords: Frequency domain methods , Functional data analysis , Locally stationary processes , spectral analysis

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 5 • October 2020
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