Estimation of local anisotropy based on level sets

Abstract

Consider an affine field $X:{\mathbb{R}}^2\to \mathbb{R}$, that is a process equal in law to $Z(A.t)$, where Z is isotropic and $A:{\mathbb{R}}^2\to {\mathbb{R}}^2$ is a linear self-adjoint transformation. The field X and transformation A will be supposed to be respectively Gaussian and definite positive. Denote $0<\mathit{\lambda }:=\frac{{\mathit{\lambda }}_2}{{\mathit{\lambda }}_1}⩽1$ the ratio of the eigenvalues of A, let ${\mathit{\lambda }}_1$, ${\mathit{\lambda }}_2$ with ${\mathit{\lambda }}_2⩽{\mathit{\lambda }}_1$. This paper is aimed at testing the null hypothesis “X is isotropic” versus the alternative “X is affine”. Roughly speaking, this amounts to testing “$\mathit{\lambda }=1$” versus “$\mathit{\lambda }<1$”. By setting level u in $\mathbb{R}$, this is implemented by the partial observations of process X through some particular level functionals viewed over a square $T_n$, which grows to ${\mathbb{R}}^2$. This leads us to provide estimators for the affinity parameters that are shown to be almost surely consistent. Their asymptotic normality results provide confidence intervals for parameters.

This paper offered an important opportunity to study general level functionals near the level u and for a fixed bounded rectangle T of ${\mathbb{R}}^2$, part of the difficulties arises from the fact that the topology of level set ${\text{C}}_{T,X}(u)=\left\{t\in T:X(t)=u\right\}$ can be irregular, even if the trajectories of X are regular. A significant part of the paper is dedicated to show the $L^2$-continuity in the level u of these general functionals.