The Dagum and auxiliary covariance families: Towards reconciling two-parameter models that separate fractal dimension and the Hurst effect

The functional statistical framework is considered to address the problem of least-squares estimation of the realizations of fractal and long-range dependence Gaussian random signals, from the observation of the corresponding response surface. The statistical methodology applied is based on the functional regression model. The geometrical properties of the separable Hilbert spaces of functions, where the response surface and the signal of interest lie, are considered for removing the ill-posed nature of the estimation problem, due to the non-locality of the integro-pseudodifferential operators involved. Specifically, the local and asymptotic properties of the spectra of fractal and long-range dependence random fields in the Linnik-type, Dagum-type and auxiliary families are analyzed to derive a stable solution to the associated functional estimation problem. Their pseudodifferential representation and Reproducing Kernel Hilbert Space (RKHS) characterization are also derived for describing the geometrical properties of the spaces where the functional random variables involved in the corresponding regression problem can be found.