## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 11, Number 1 (2017), 843-890.

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

Elena Di Bernardino, Anne Estrade, and José R. León

#### 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.

#### Article information

**Source**

Electron. J. Statist., Volume 11, Number 1 (2017), 843-890.

**Dates**

Received: July 2016

First available in Project Euclid: 28 March 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1490688316

**Digital Object Identifier**

doi:10.1214/17-EJS1248

**Mathematical Reviews number (MathSciNet)**

MR3629017

**Zentralblatt MATH identifier**

1362.62098

**Subjects**

Primary: 62G10: Hypothesis testing

Secondary: 60G10: Stationary processes 60G15: Gaussian processes 60G60: Random fields

**Keywords**

Test of Gaussianity Gaussian fields excursion sets level sets Euler characteristic crossings

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Di Bernardino, Elena; Estrade, Anne; León, José R. A test of Gaussianity based on the Euler characteristic of excursion sets. Electron. J. Statist. 11 (2017), no. 1, 843--890. doi:10.1214/17-EJS1248. https://projecteuclid.org/euclid.ejs/1490688316