THE SLEX MODEL OF A NON-STATIONARY RANDOM PROCESS

HERNANDO OMBAO^{1}, JONATHAN RAZ^{2}, RAINER VON SACHS^{3} AND WENSHENG GUO^{2}

1Departments of Statistics and Psychiatry, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.

2Division of Biostatistics, University of Pennsylvania, Philadelphia, PA 19104, U.S.A. 3Institut de Statistique, Université Catholique de Louvain, Voie du Roman Pays 20, B-1348,

Louvain-la-Neuve, Belgium

(Received October 15, 2000; revised July 3, 2001)

Abstract. We propose a new model for non-stationary random processes to represent time series with a time-varying spectral structure. Our SLEX model can be considered as a discrete time-dependent Cramér spectral representation. It is based on the so-called Smooth Localized complex EXponential basis functions which are orthogonal and localized in both time and frequency domains. Our model delivers a finite sample size representation of a SLEX process having a SLEX spectrum which is piecewise constant over time segments. In addition, we embed it into a sequence of models with a limit spectrum, a smoothly in time varying “evolutionary” spectrum. Hence, we develop the SLEX model parallel to the Dahlhaus (1997, Ann. Statist., 25, 1–37) model of local stationarity, and we show that the two models are asymptotically mean square equivalent. Moreover, to define both the growing complexity of our model sequence and the regularity of the SLEX spectrum we use a wavelet expansion of the spectrum over time. Finally, we develop theory on how to estimate the spectral quantities, and we brieΩy discuss how to form inference based on resampling (bootstrapping) made possible by the special structure of the SLEX model which allows for simple synthesis of non-stationary processes.

Key words and phrases: Bootstrap, Fourier functions, Haar wavelet representation, locally stationary time series, periodograms, SLEX functions, spectral estimation, stationary time series.