A nonparametric test for stationarity in functional time series

摘要

We propose a new measure for stationarity of a functional time series, whichis based on an explicit representation of the $L^2$-distance between thespectral density operator of a non-stationary process and its best($L^2$-)approximation by a spectral density operator corresponding to astationary process. This distance can easily be estimated by sums ofHilbert-Schmidt inner products of periodogram operators (evaluated at differentfrequencies), and asymptotic normality of an appropriately standardised versionof the estimator can be established for the corresponding estimate under thenull hypothesis and alternative. As a result we obtain confidence intervals forthe discrepancy of the underlying process from a functional stationary processand a simple asymptotic frequency domain level $lpha$ test (using thequantiles of the normal distribution) for the hypothesis of stationarity offunctional time series. Moreover, the new methodology allows also to testprecise hypotheses of the form "the functional time series is approximatelystationarity", which means that the new measure of stationarity is smaller thana given threshold. Thus in contrast to methods proposed in the literature ourapproach also allows to test for "relevant" deviations from stationarity. Wedemonstrate in a small simulation study that the new method has very goodfinite sample properties and compare it with the currently availablealternative procedures. Moreover, we apply our test on annual temperaturecurves.