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 and antireflective filters. The associated smoothing matrices, though non-symmetric, have analytically known spectral decomposition. 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.