A wavelet-based test for stationarity

We develop a test for stationarity of a time series against the alternative of a time-varying covariance structure. Using localized versions of the periodogram, we obtain empirical versions of a reasonable notion of a time-varying spectral density. Coeecients w.r.t. a Haar wavelet series expansion of such a time-varying periodogram are an indicator whether there is some deviation from covariance stationarity. We propose a test based on the limit distribution of these empirical coeecients. 1 1 1. Introduction The simplifying assumption of stationarity, i.e. a second{order structure which is constant over time, is very often made in time series analysis. If this is actually adequate, it essentially simpliies the statistical analysis and allows one to use classical methods of data analysis. However, in the case of a strong deviation from stationarity, e.g. sudden or periodic changes in the covariance, this erroneous assumption can cause unexpected eeects, and can nally lead to wrong conclusions about the underlying process. Examples of nonstationary processes are numerous, and can be found, for instance, in biomedical time series analysis whether the measurements are of blood pressure, enzyme levels, biomechanical movements or heartbeats, etc. In particular, in the analysis of series with pulsatile components, often a classical model of stationarity (and Gaussianity) is not suucient to explain the data. We like to refer to Normolle and Brown (1994) who, among others, used a series of luteinizing hormone concentrations to illustrate such phenomena, occurring in the detection of seasonalities in the series. Other examples of nonstationary phenomena derive from electrical and acoustical engineering (Doppler signals, speech analysis, EEG's and ECG's), geophysics and economics. Therefore, it is important to have some guidelines to assess the adequacy of the assumption of stationarity. When we focus on the mean and the covariance structure as the central characteristics of a time series, we can readily get some impression about possible deviations from stationarity by looking at nonparametric estimates of the mean function m and the time-varying spectral density. The latter has been treated by Neumann and von Sachs (1997), whereas von Sachs and MacGibbon (1997) is an example for estimation of local variation in the trend of the data (e.g., again the luteinizing hormone data) in the presence of a time{changing stochastic uctuation. If one is interested in a decision rule on a more formal level, then one may employ tests for the hypothesis of stationarity. There are diierent kinds of deviations …