The Annals of Statistics

Consistent estimates of deformed isotropic Gaussian random fields on the plane

Ethan Anderes and Sourav Chatterjee

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Abstract

This paper proves fixed domain asymptotic results for estimating a smooth invertible transformation f: ℝ2→ℝ2 when observing the deformed random field Zf on a dense grid in a bounded, simply connected domain Ω, where Z is assumed to be an isotropic Gaussian random field on ℝ2. The estimate is constructed on a simply connected domain U, such that ⊂Ω 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 Rθf+c uniformly on compact subsets of U with probability one as the grid spacing goes to zero, where Rθ is an unidentifiable rotation and c is an unidentifiable translation.

Article information

Source
Ann. Statist., Volume 37, Number 5A (2009), 2324-2350.

Dates
First available in Project Euclid: 15 July 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1247663757

Digital Object Identifier
doi:10.1214/08-AOS647

Mathematical Reviews number (MathSciNet)
MR2543694

Zentralblatt MATH identifier
1171.62056

Subjects
Primary: 60G60: Random fields 62M30: Spatial processes 62M40: Random fields; image analysis
Secondary: 62G05: Estimation

Keywords
Deformation quasiconformal maps nonstationary random fields Bergman space

Citation

Anderes, Ethan; Chatterjee, Sourav. Consistent estimates of deformed isotropic Gaussian random fields on the plane. Ann. Statist. 37 (2009), no. 5A, 2324--2350. doi:10.1214/08-AOS647. https://projecteuclid.org/euclid.aos/1247663757


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