Whittle estimation with (quasi-)analytic wavelets

The notion of long-memory is considered in the case of multivariate time series, not necessarily Gaussian nor stationary. The long-memory characteristics are defined by the long-memory parameters describing the autocorrelation structure of each process and the long-run covariance measuring the coupling between time series. A phase term is present in the model to widen the classes of models. We introduce a representation of the time series by quasi-analytic wavelets for inference in this setting. We first show that the covariance of the wavelet coefficients provides an adequate estimator of the covariance structure of the processes, including the phase term. Consistent estimators are then proposed which is based on a Whittle approximation. Simulations highlight a satisfactory behavior of the estimation on finite samples on some 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.