Abstract
In the present paper we study three geometrical characteristics for the excursion sets of a two-dimensional stationary isotropic random field. First, we show that these characteristics can be estimated without bias if the considered field satisfies a kinematic formula, this is for instance the case of fields given by a function of smooth Gaussian fields or of some shot noise fields. By using the proposed estimators of these geometric characteristics, we describe some inference procedures for the estimation of the parameters of the field. An extensive simulation study illustrates the performances of each estimator. Then, we use the Euler characteristic estimator to build a test to determine whether a given field is Gaussian or not, when compared to various alternatives. The test is based on a sparse information, i.e., the excursion sets for two different levels of the field to be tested. Finally, the proposed test is adapted to an applied case, synthesized 2D digital mammograms.
Highlights
In this paper we are interested in using simple geometrical objects to build inference and testing procedures to recover global characteristics of a twodimensional stationary isotropic random field by using a sparse information
Σ−u11,/u22 is bounded away from 0 and if a joint central limit theorem for Ci/T (X, u1), Ci/T (X, u2), i = 0, 1 for X a Student fields was known this would entail the consistency of the test, i.e., PH1(φT (N)
We have presented new statistical tools for inferring parameters and testing Gaussianity when only a sparse observation of a 2D random field is available, namely only the excursion set(s) above one or two level(s) within a large window
Summary
In this paper we are interested in using simple geometrical objects to build inference and testing procedures to recover global characteristics of a twodimensional stationary isotropic random field by using a sparse information. With the same objective -joint study of all Minkowski functionals-, but in another context -excursion sets of random fields defined on a discrete space, one can quote the recent papers [29] and [13]. The estimator of the LK curvature corresponding to the Euler characteristic allows us to construct a test to determine if two observed excursion sets result from a Gaussian field or not. The strength of this test is that it is independent of a specific choice of the correlation function of the considered random field, in particular it does not depend on its second spectral moment.
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