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October, 1995 Periodogram-Based Estimators of Fractal Properties
Grace Chan, Peter Hall, D. S. Poskitt
Ann. Statist. 23(5): 1684-1711 (October, 1995). DOI: 10.1214/aos/1176324319


We suggest an estimator, based on the periodogram, of the fractal index and fractal dimension of a continuous, stationary Gaussian process. We argue that the cosine part of the periodogram is more appropriate than the full periodogram for this application. The term "semiperiodogram" is used to describe the cosine component, and our estimator is based on simple linear regression of the logarithm of the semiperiodogram on the algorithm of frequency. Theoretical properties of the estimator, including its bias, variance and asymptotic distribution, are derived. Consistency is possible using only a small trace of the process, recorded over a fixed interval. We do not need to model the covariance function parametrically, and assume only mild conditions on the behaviour of the covariance in the neighbourhood of the origin. The issue of aliasing is discussed in both theoretical and numerical terms, and the numerical properties of the estimator are assessed in a simulation study.


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Grace Chan. Peter Hall. D. S. Poskitt. "Periodogram-Based Estimators of Fractal Properties." Ann. Statist. 23 (5) 1684 - 1711, October, 1995.


Published: October, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0843.62090
MathSciNet: MR1370303
Digital Object Identifier: 10.1214/aos/1176324319

Primary: 62M10
Secondary: 62E10 , 62G05

Keywords: Aliasing , bias , Covariance , fractal dimension , fractal index , frequency , Gaussian process , periodogram , regression , self-similar process , semiperiodogram

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 5 • October, 1995
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