A test of Gaussianity based on the Euler characteristic of excursion sets

Abstract

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.