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June 2004 Estimation of fractal dimension for a class of non-Gaussian stationary processes and fields
Grace Chan, Andrew T. A. Wood
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Ann. Statist. 32(3): 1222-1260 (June 2004). DOI: 10.1214/009053604000000346

Abstract

We present the asymptotic distribution theory for a class of increment-based estimators of the fractal dimension of a random field of the form g{X(t)}, where g:RR is an unknown smooth function and X(t) is a real-valued stationary Gaussian field on Rd, d=1 or 2, whose covariance function obeys a power law at the origin. The relevant theoretical framework here is “fixed domain” (or “infill”) asymptotics. Surprisingly, the limit theory in this non-Gaussian case is somewhat richer than in the Gaussian case (the latter is recovered when g is affine), in part because estimators of the type considered may have an asymptotic variance which is random in the limit. Broadly, when g is smooth and nonaffine, three types of limit distributions can arise, types (i), (ii) and (iii), say. Each type can be represented as a random integral. More specifically, type (i) can be represented as the integral of a certain random function with respect to Lebesgue measure; type (ii) can be represented as the integral of a second random function with respect to an independent Gaussian random measure; and type (iii) can be represented as a Wiener–Itô integral of order 2. Which type occurs depends on a combination of the following factors: the roughness of X(t), whether d=1 or d=2 and the order of the increment which is used. Another notable feature of our results is that, even though the estimators we consider are based on a variogram, no moment conditions are required on the observed field g{X(t)} for the limit theory to hold. The results of a numerical study are also presented.

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Grace Chan. Andrew T. A. Wood. "Estimation of fractal dimension for a class of non-Gaussian stationary processes and fields." Ann. Statist. 32 (3) 1222 - 1260, June 2004. https://doi.org/10.1214/009053604000000346

Information

Published: June 2004
First available in Project Euclid: 24 May 2004

zbMATH: 1046.62103
MathSciNet: MR2065204
Digital Object Identifier: 10.1214/009053604000000346

Subjects:
Primary: 62M99
Secondary: 62E20

Keywords: Gaussian random measure , Hermite polynomial , increments , stationary random field , variogram , Wiener–Itô integral

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 3 • June 2004
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