Open Access
2019 Lipschitz-Killing curvatures of excursion sets for two-dimensional random fields
Hermine Biermé, Elena Di Bernardino, Céline Duval, Anne Estrade
Electron. J. Statist. 13(1): 536-581 (2019). DOI: 10.1214/19-EJS1530


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.


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Hermine Biermé. Elena Di Bernardino. Céline Duval. Anne Estrade. "Lipschitz-Killing curvatures of excursion sets for two-dimensional random fields." Electron. J. Statist. 13 (1) 536 - 581, 2019.


Received: 1 May 2018; Published: 2019
First available in Project Euclid: 14 February 2019

zbMATH: 1406.60076
MathSciNet: MR3911693
Digital Object Identifier: 10.1214/19-EJS1530

Primary: 60G60 , 62F12
Secondary: 60G10 , 62F03

Keywords: crossings , Euler characteristic , Excursion sets , Gaussian and shot-noise fields , Level sets , Test of Gaussianity

Vol.13 • No. 1 • 2019
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