Spectral filtering for trend estimation

This paper deals with trend estimation at the boundaries of a time se-ries by means of smoothing methods. After deriving the asymptotic properties ofsequences of matrices associated with linear smoothers, two classes of asymmet-ric filters that approximate a given symmetric estimator are introduced: the reflec-tive filters (RF) and antireflective filters (AF). The associated smoothing matrices,though non-symmetric, have analytically known spectral decomposition. The paperanalyses the properties of the new filters and considers RF and AF algebras for ap-proximating the eigensystems of time series smoothing matrices. A strategy for aspectral filter design is also discussed.