The Annals of Statistics

Periodogram-Based Estimators of Fractal Properties

Grace Chan, Peter Hall, and D. S. Poskitt

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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.

Article information

Ann. Statist. Volume 23, Number 5 (1995), 1684-1711.

First available in Project Euclid: 11 April 2007

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Zentralblatt MATH identifier


Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62G05: Estimation 62E10: Characterization and structure theory

Aliasing bias covariance fractal dimension fractal index frequency Gaussian process periodogram regression self-similar process semiperiodogram


Chan, Grace; Hall, Peter; Poskitt, D. S. Periodogram-Based Estimators of Fractal Properties. Ann. Statist. 23 (1995), no. 5, 1684--1711. doi:10.1214/aos/1176324319.

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