Consistent estimates of deformed isotropic Gaussian random fields on the plane

Abstract

This paper proves fixed domain asymptotic results for estimating a smooth invertible transformation $f:{\mathbb{R}}^2\to {\mathbb{R}}^2$ when observing the deformed random field $Z\circ f$ on a dense grid in a bounded, simply connected domain $\Omega$, where $Z$ is assumed to be an isotropic Gaussian random field on ${\mathbb{R}}^2$. The estimate $\hat{f}$ is constructed on a simply connected domain $U$, such that $\overline{U}\subset \Omega$ and is defined using kernel smoothed quadratic variations, Bergman projections and results from quasiconformal theory. We show, under mild assumptions on the random field $Z$ and the deformation $f$, that $\hat{f}\to R_{\theta }f+c$ uniformly on compact subsets of $U$ with probability one as the grid spacing goes to zero, where $R_{\theta }$ is an unidentifiable rotation and $c$ is an unidentifiable translation.

Published in at http://dx.doi.org/10.1214/08-AOS647 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Estimating deformations of isotropic Gaussian random fields on the plane
• Consistent estimates of deformed isotropic Gaussian random fields on the plane

Cited by in corpus (1)

• Consistent estimates of deformed isotropic Gaussian random fields on the plane