Research Article

Estimation of anisotropic Gaussian fields through Radon transform

Biermé, Hermine^a1 and Richard, Frédéric^a1

^a1 MAP5-UMR 8145, Université René Descartes45, rue des Saints-Pères, 75270 Paris cedex 06 France, hermine.bierme@math-info.univ-paris5.fr; frederic.richard@math-info.univ-paris5.fr

Abstract

We estimate the anisotropic index of an anisotropic fractional Brownian field. For all directions, we give a convergent estimator of the value of the anisotropic index in this direction, based on generalized quadratic variations. We also prove a central limit theorem. First we present a result of identification that relies on the asymptotic behavior of the spectral density of a process. Then, we define Radon transforms of the anisotropic fractional Brownian field and prove that these processes admit a spectral density satisfying the previous assumptions. Finally we use simulated fields to test the proposed estimator in different anisotropic and isotropic cases. Results show that the estimator behaves similarly in all cases and is able to detect anisotropy quite accurately.

(Received February 15 2006)

(Revised December 12 2006)

(Online publication November 13 2007)

Key Words:

• Anisotropic Gaussian fields;
• identification;
• estimator;
• asymptotic normality;
• Radon transform.

Mathematics Subject Classification:

• 60G60;
• 62M40;
• 60G15;
• 60G10;
• 60G17;
• 60G35;
• 44A12