The paper concerns the design of nonparametric low-pass filters that have the property of reproducing a polynomial of a given degree. Two approaches are considered. The first is locally weighted polynomial regression (LWPR), which leads to linear filters depending on three parameters: the...

This note is concerned with the spectral properties of matrices associated with linear smoothers. We derive analytical results on the eigenvalues and eigenvectors of smoothing matrices by interpreting the latter as perturbations of matrices belonging to algebras with known spectral properties,...

A univariate first order stochastic cycle can be represented as an element of a bivariate first order vector autoregressive process, or VAR(1), where the transition matrix is associated with a Givens rotation. From the geometrical viewpoint, the kernel of the cyclical dynamics is described by a...

The paper establishes the conditions under which the generalised least squares estimator of the regression parameters is equivalent to the weighted least squares estimator. The equivalence conditions have interesting applications in local polynomial regression and kernel smoothing. Specifically,...

The variance profile is defined as the power mean of the spectral density function of a stationary stochastic process. It is a continuous and non-decreasing function of the power parameter, p, which returns the minimum of the spectrum (p → −∞), the interpolation error variance (harmonic...

The exponential model for the spectrum of a time series and its fractional extensions are based on the Fourier series expansion of the logarithm of the spectral density. The coefficients of the expansion form the cepstrum of the time series. After deriving the cepstrum of important classes of...