Spectral filtering for trend estimation

This paper deals with trend estimation at the boundaries of a time series by means of smoothing methods. After deriving the asymptotic properties of sequences of matrices associated with linear smoothers, two classes of asym- metric filters that approximate a given symmetric estimator are introduced: the reflective filters and antireflective filters. The associated smoothing ma- trices, though non-symmetric, have analytically known spectral decompo- sition. The paper analyses the properties of the new filters and considers reflective and antireflective algebras for approximating the eigensystems of time series smoothing matrices. Based on this, a thresholding strategy for a spectral filter design is discussed.