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

Elena Di Bernardino (1) , Anne Estrade (2) , José Rafael León (3)

1 CEDRIC - Centre d'études et de recherche en informatique et communications
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
3 UCV - Universidad Central de Venezuela

In the present paper, we deal with a stationary random field $X:\R^d \to \R$ 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 some 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 $u$ 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. Simulations are performed both in one dimensional case ($d=1$) and in two dimensional case ($d=2$), using R.

Mots clés: Test of Gaussianity; Gaussian fields; Excursion sets; Euler characteristic; Crossings

Domaines: Statistiques [stat]; Probabilités [math.PR]

Elena Di Bernardino, Anne Estrade, José Rafael León. A test of Gaussianity based on the Euler characteristic of excursion sets. 2016. ⟨hal-01345225v1⟩