The Annals of Statistics

Periodogram-Based Estimators of Fractal Properties

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

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 11 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1176324319

Digital Object Identifier
doi:10.1214/aos/1176324319

Mathematical Reviews number (MathSciNet)
MR1370303

Zentralblatt MATH identifier
0843.62090

JSTOR
links.jstor.org

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

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

Citation

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. http://projecteuclid.org/euclid.aos/1176324319.


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