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Title: A nonparametric test for stationarity in functional time series Authors:  Anne van Delft - Columbia University (United States) [presenting]
Pramita Bagchi - Ruhr University Bochum (Germany)
Vaidotas Characiejus - University of California, Davis (United States)
Holger Dette - Ruhr-Universitaet Bochum (Germany)
Abstract: A new measure for stationarity of a functional time series is proposed, which is based on an explicit representation of the L2-distance between the spectral density operator of a non-stationary process and its best L2-approximation by a spectral density operator corresponding to a stationary process. This distance can easily be estimated by sums of Hilbert Schmidt inner products of periodogram operators (evaluated at different frequencies), and asymptotic normality of an appropriately standardised version of the estimator can be established for the corresponding estimate under the null hypothesis and alternative. As a result, we obtain condence intervals for the discrepancy of the underlying process from a functional stationary process and a simple asymptotic frequency domain level alpha test (using the quantiles of the normal distribution) for the hypothesis of stationarity of functional time series. Moreover, the new methodology allows also to test precise hypotheses of the form the functional time series is approximately stationarity, which means that the new measure of stationarity is smaller than a given threshold. Our approach therefore also allows to test for relevant deviations from stationarity.