Adaptive wavelet decompositions of stationary time series‡

A general and flexible framework for the wavelet‐based decompositions of stationary time series in discrete time, called adaptive wavelet decompositions (AWDs), is introduced. It is shown that several particular AWDs can be constructed with the aim of providing decomposition (approximation and detail) coefficients that exhibit certain nice statistical properties, where the latter can be chosen based on a range of theoretical or applied considerations. AWDs make use of a Fast Wavelet Transform‐like algorithm whose filters ‐ in contrast with their counterparts in Orthogonal Wavelet Decompositions (OWDs) – may depend on the scale. As with OWDs, this algorithm has good properties such as computational efficiency and invariance to polynomial trends. A property whose pursuit plays a central role in this work is the decorrelation of the detail coefficients. For many time series models (e.g., FARIMA(0,δ,0)), the AWD filters can be defined so that the resulting AWD detail coefficients are all (exactly) decorrelated. The corresponding AWDs, called Exact AWDs (EAWDs), are particularly useful in simulation of Gaussian stationary time series, if the associated filters have a fast decay. The proposed simulation methods generalize and improve upon existing wavelet‐based ones. AWDs for which the detail coefficients are not exactly decorrelated, but still more decorrelated than those of OWDs, are referred to as approximate AWDs (AAWDs). They can be obtained by truncating EAWD filters, or by adopting some of the existing approaches to modeling the dependence structure of the OWD detail coefficients (e.g., Craigmile et al., 2005). AAWDs naturally lead to new wavelet‐based Maximum Likelihood estimators. The performance of these estimators is investigated through simulations and from some theoretical standpoints. The focus in estimation is also on Gaussian stationary series, though the method is expected to work for non‐Gaussian stationary series as well.