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 asymmetric filters that approximate a given symmetric estimator are introduced: the reflective filters (RF) and antireflective filters (AF). The associated smoothing matrices, though non-symmetric, have analytically known spectral decomposition. The paper analyses the properties of the new filters and considers RF and AF algebras for approximating the eigensystems of time series smoothing matrices. A strategy for a spectral filter design is also discussed.