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 func...

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Bibliographic Details
Published in:Probabilistic engineering mechanics 2011-04, Vol.26 (2), p.259-268
Main Authors: Ruiz-Medina, Maria Dolores, Porcu, Emilio, Fernandez-Pascual, Rosaura
Format: Article
Language:English
Subjects:
Dagum family
Exact sciences and technology
Experimental design
Fractal analysis
Fractals
Functional regression
Fundamental areas of phenomenology (including applications)
Hausdorff dimension
Hilbert space
Hurst index
Ill posed problems
Linear inference, regression
Mathematical models
Mathematics
Mean-square Hölder exponent
Physics
Probability and statistics
Regression
Representations
Response surface
Response surfaces
RKHS
Sciences and techniques of general use
Sobolev spaces
Solid mechanics
Spectra
Static elasticity (thermoelasticity...)
Statistical Volume Element
Statistics
Structural and continuum mechanics
Thermoelasticity
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Summary: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.
ISSN:0266-8920
1878-4275
DOI:10.1016/j.probengmech.2010.08.002