Statistica Sinica 26 (2016), 313-357
doi:http://dx.doi.org/10.5705/ss.2014.039

MEASURING STATIONARITY IN LONG-MEMORY
PROCESSES
Kemal Sen, Philip Preuß and Holger Dette
Ruhr-Universität Bochum

Abstract: In this paper we consider the problem of measuring stationarity in locally stationary long-memory processes. We introduce an L2-distance between the spectral density of the locally stationary process and its best approximation under the assumption of stationarity. The distance is estimated by a numerical approximation of the integrated spectral periodogram and asymptotic normality of the resulting estimate is established. The results can be used to construct a simple test for the hypothesis of stationarity in locally stationary long-range dependent processes. We also propose a bootstrap procedure to improve the approximation of the nominal level and prove its consistency. Throughout the paper, we work with Riemann sums of a squared periodogram instead of integrals (as it is usually done in the literature) and as a by-product of independent interest it is demonstrated that the two approaches behave differently in the limit.

Key words and phrases: Bootstrap, empirical spectral measure, goodness-of-fit tests, integrated periodogram, locally stationary process, long-memory, non-stationary processes, spectral density.