Modelling and Estimation of the Time{varying Structure of Nonstationary Time Series

Recently, quite a few new statistical models have been added to the variety of existing ones which try to capture the time{varying structure of nonstationary stochastic processes. Examples are numerous and can be found, e.g., in economics, speech and sound processing and the biomedical sciences. This paper gives an overview of diierent modelling approaches, e.g. locally stationary time{frequency representations or local time{scale decompositions. It also outlines some of the statistical procedures that estimate characteristic second-order quantities of these stochastic processes, e.g. local autocovariances or local spectral densities. Among those, we will nd spectograms, Wigner{Ville estimators, localized periodograms as estimates of evolutionary spectra, methods based on local cosine packets and wavelet spectrum estimators. We discuss consistent estimation and adaption of the estimators to the actual time{frequency or time{scale content of the time series and also present some data examples.