Whittle estimation with (quasi-)analytic wavelets

In the general setting of long-memory multivariate time series, the long-memory characteristics are defined by two components. The long-memory parameters describe the autocorrelation of each time series. And the long-run covariance measures the coupling between time series, with general phase parameters. This wide class of models provides time series not necessarily Gaussian nor stationary. It is of interest to estimate the parameters: long-memory, long-run covariance and general phase. This inference is not possible using real wavelets decomposition or Fourier analysis. Our purpose is to define an inference approach based on a representation using quasi-analytic wavelets. We first show that the covariance of the wavelet coefficients provides an adequate estimator of the covariance structure including the phase term. Consistent estimators based on a Whittle approximation are then proposed. Simulations highlight a satisfactory behavior of the estimation on finite samples on linear time series and on multivariate fractional Brownian motions. An application on a real dataset in neuroscience is displayed, where long-memory and brain connectivity are inferred.