Abstract: In this paper, we propose a nonparametric method to estimate the spectrum of a multivariate locally stationary process. The time-varying spectrum is assumed to be smooth in both time and frequency. In order to ensure that the final estimate of the multivariate spectrum is positive definite while allowing enough flexibility in estimating each of its elements, we propose to smooth the Cholesky decomposition of an initial spectral estimate and to reconstruct the final spectral estimate from the smoothed Cholesky elements. We propose a two-stage estimation procedure. The first stage approximates the locally stationary process by a piecewise stationary time series to obtain the local estimate of the time varying spectrum and its Cholesky decomposition on discrete time-frequency points. The second stage uses a smoothing spline ANOVA to jointly smooth each Cholesky element in both time and frequency, and reconstructs the final estimate of the time varying multivariate spectrum for any time-frequency point. The final estimate is a smooth function in time and frequency, has a global interpretation, and is consistent and positive definite. We show that the Cholesky decomposition of a time varying spectrum can be used as a transfer function to generate a locally stationary time series with the designed spectrum. This not only provides us flexibility in simulations, but also allows us to construct bootstrap confidence intervals on the time varying multivariate spectrum. A simulation is conducted to investigate its performance and an application to an EEG data set recorded during an epileptic seizure is used as an illustration.
Key words and phrases: Bootstrap, Cholesky decomposition, locally stationary time series, smoothing spline, spectral analysis.