Analyzing and modeling long-memory time series using fractional spline wavelets - Pinto, Mateus Gonzalez de Freitas (2024)
Fractional splines extend Schoenberg\'s B-splines to fractional orders, which have been shown to fulfill all the requirements to form wavelet bases. Nevertheless, some of these fractional spline wavelets act as fractional difference operators for signals with essentially low-pass behavior and with a pole around the origin, making them useful in the analysis of series with fractal behavior. Using the fact that this family of wavelets acts approximately as a fractional difference operator in the Fourier domain, this thesis proposes two novel estimators for the long-memory parameter of a time series based on the fractional spline discrete wavelet transform (FrDWT), one heuristic and the other based on maximum likelihood. In this thesis, we demonstrate the fractional differentiation properties of fractional spline wavelets, as well as a theorem that allows for the construction of a procedure for whitening fractional noises. Simulations and examples are provided to illustrate the proposed methods, verifying their competitiveness with other proposals in the literature. Finally, we present the behavior of the proposed estimator on real data, verifying its dominance over other widely employed methods in the time series literature.