Open Access
2017 A test of Gaussianity based on the Euler characteristic of excursion sets
Elena Di Bernardino, Anne Estrade, José R. León
Electron. J. Statist. 11(1): 843-890 (2017). DOI: 10.1214/17-EJS1248


In the present paper, we deal with a stationary isotropic random field $X:{\mathbb{R}}^{d}\to{\mathbb{R}}$ and we assume it is partially observed through some level functionals. We aim at providing a methodology for a test of Gaussianity based on this information. More precisely, the level functionals are given by the Euler characteristic of the excursion sets above a finite number of levels. On the one hand, we study the properties of these level functionals under the hypothesis that the random field $X$ is Gaussian. In particular, we focus on the mapping that associates to any level $u$ the expected Euler characteristic of the excursion set above level $u$. On the other hand, we study the same level functionals under alternative distributions of $X$, such as chi-square, harmonic oscillator and shot noise. In order to validate our methodology, a part of the work consists in numerical experimentations. We generate Monte-Carlo samples of Gaussian and non-Gaussian random fields and compare, from a statistical point of view, their level functionals. Goodness-of-fit $p-$values are displayed for both cases. Simulations are performed in one dimensional case ($d=1$) and in two dimensional case ($d=2$), using R.


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Elena Di Bernardino. Anne Estrade. José R. León. "A test of Gaussianity based on the Euler characteristic of excursion sets." Electron. J. Statist. 11 (1) 843 - 890, 2017.


Received: 1 July 2016; Published: 2017
First available in Project Euclid: 28 March 2017

zbMATH: 1362.62098
MathSciNet: MR3629017
Digital Object Identifier: 10.1214/17-EJS1248

Primary: 62G10
Secondary: 60G10 , 60G15 , 60G60

Keywords: crossings , Euler characteristic , Excursion sets , Gaussian fields , Level sets , Test of Gaussianity

Vol.11 • No. 1 • 2017
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