Lipschitz-Killing curvatures of excursion sets for two-dimensional random fields

Hermine Biermé (1) , Elena Di Bernardino (2) , Céline Duval (3) , Anne Estrade (3)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 7348]
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

In the present paper we study three geometrical characteristics for the excursion sets of a two-dimensional stationary isotropic random field. First, we show that these characteristics can be estimated without bias if the considered field satisfies a kinematic formula, this is for instance the case of fields given by a function of smooth Gaussian fields or of some shot noise fields. By using the proposed estimators of these geometric characteristics, we describe some inference procedures for the estimation of the parameters of the field. An extensive simulation study illustrates the performances of each estimator. Then, we use the Euler characteristic estimator to build a test to determine whether a given field is Gaussian or not, when compared to various alternatives. The test is based on a sparse information, i.e., the excursion sets for two different levels of the field to be tested. Finally, the proposed test is adapted to an applied case, synthesized 2D digital mammograms.

Mots clés: MSC 2010 subject classifications: Primary 60G60; 62F12; Secondary 62F03; 60G10 Test of Gaussianity; Gaussian and shot-noise fields; excursion sets; level sets; Euler characteristic; crossings

Domaines: Statistiques [stat]

Hermine Biermé, Elena Di Bernardino, Céline Duval, Anne Estrade. Lipschitz-Killing curvatures of excursion sets for two-dimensional random fields. Electronic Journal of Statistics , 2019, 13, pp.536 - 581. ⟨10.1214/19-ejs1530⟩. ⟨hal-03080411⟩