Abstract
In the present paper, we deal with a stationary isotropic random field X : R d → 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 func-tionals 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.
Highlights
The question of the Gaussianity of a phenomenon is a historical and fundamental problem in statistical literature
We have proposed a methodology to test whether a random field defined on Rd is Gaussian
The test statistics are computed from the observation of a single realisation, precisely the Euler characteristic of excursion sets at moderate levels is concerned
Summary
The question of the Gaussianity of a phenomenon is a historical and fundamental problem in statistical literature. If the underlying standard random field X is not Gaussian with a given second spectral moment (in particular if X is a χ2 or a Kramer oscillator process), we deliberately center again, as in the I. case, the Zki ’s variables by using the (wrong) theoretical Gaussian mean of excursions EC at level uk (see Equation (6)) with the same second spectral moment In this case, we obtain a very small goodness-of-fit p−values for the chi-squared distribution associated to the considered test statistics based on Zki ’s (see Sections 4.2.1 and 4.2.2). In both cases, we give an explicit formula for the mean EC of excursion sets. In order to keep the methodological spirit of the present paper, we have reported the technical proofs to the appendix section
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