Statistics for Gaussian random fields with unknown location and scale using Lipschitz‐Killing curvatures

Abstract

In the present article we study the average of Lipschitz‐Killing (LK) curvatures of the excursion set of a stationary isotropic Gaussian field X on ℝ2. The novelty is that the field can be nonstandard, that is, with unknown mean and variance, which is more realistic from an applied viewpoint. To cope with the unknown location and scale parameters of X, we introduce novel fundamental quantities called effective level and effective spectral moment. We propose unbiased and asymptotically normal estimators of these parameters. From these asymptotic results, we build a test to determine if two images of excursion sets can be compared. This test is applied on both synthesized and real mammograms. Meanwhile, we establish the consistency of the empirical variance estimators of the third LK curvature under a weak condition on the correlation function of X.

On the perimeter estimation of pixelated excursion sets of two‐dimensional anisotropic random fields

“…Suppose that there exists a positive-definite matrix A such that the random field X is equal in distribution to {Y (As) ∶ s ∈ R 2 }, for some C 2 , stationary, isotropic, centered, Gaussian random field Y with covariance function r(h), h ∈ R The proof can be found in Section 5. We remark that a vast literature exists on the asymptotic distribution of level functionals of Gaussian random fields (Beliaev et al, 2020;Di Bernardino et al, 2017;Di Bernardino & Duval, 2022;Meschenmoser & Shashkin, 2013;Shashkin, 2013;Wschebor, 1985), in which case, the asymptotic variance-covariance matrix in ( 11) can be written by projecting the Gaussian functionals of interest onto the Itô-Wiener chaos (the interested reader is referred, for instance, to Kratz & León, 2001;Estrade & León, 2016;Müller, 2017;Kratz & Vadlamani, 2018;Berzin, 2021).…”

“…LKCs have recently been used to create several statistical procedures including parametric inference (Biermé et al, 2019; Di Bernardino & Duval, 2022) and tests of Gaussianity (Di Bernardino et al, 2017), isotropy (Berzin, 2021; Cabaña, 1987; Fournier, 2018), and symmetry of marginal distributions the underlying fields (Abaach et al, 2021). Di Bernardino et al (2020) quantify perturbation via the LKCs and provides a quantitative non‐Gaussian limit theorem of the perturbed excursion area behavior.…”

Spatial extremes and stochastic geometry for Gaussian-based peaks-over-threshold processes

“…Central-limit theorems for LKCs were proposed by [43] and [51]. The setting of a stationary isotropic Gaussian field on R 2 with unknown mean and variance was recently studied in [28]. The previously cited statistical results permit to derive inference procedures (see, for instance, [45,12,28]) and to test for isotropy, Gaussianity, and marginal symmetry of the underlying fields (see, for instance, [16,9,27,12,1,36]).…”

“…The setting of a stationary isotropic Gaussian field on R 2 with unknown mean and variance was recently studied in [28]. The previously cited statistical results permit to derive inference procedures (see, for instance, [45,12,28]) and to test for isotropy, Gaussianity, and marginal symmetry of the underlying fields (see, for instance, [16,9,27,12,1,36]). [28] propose a test to determine if two images of excursion sets can be compared based on consistent estimators of LKCs.…”

“…The previously cited statistical results permit to derive inference procedures (see, for instance, [45,12,28]) and to test for isotropy, Gaussianity, and marginal symmetry of the underlying fields (see, for instance, [16,9,27,12,1,36]). [28] propose a test to determine if two images of excursion sets can be compared based on consistent estimators of LKCs. Notice that previous inference methods and tests based on LKCs only require observation of one or two excursion sets, whereas efficient inference about the covariance function and the marginal distribution of X requires more complete observation of the entire field; see [56,52], for instance.…”