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Zbl 0843.62090
Chan, Grace; Hall, Peter; Poskitt, D.S.
Periodogram-based estimators of fractal properties. (English)
[J] Ann. Stat. 23, No.5, 1684-1711 (1995). ISSN 0090-5364

Summary: 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.

MSC 2000:: *62M10 Time series, etc. (statistics)
62G05 Nonparametric estimation
62M15 Spectral analysis of processes
65C99 Numerical simulation

Keywords: semiperiodogram; consistency; self-similar process; periodogram; fractal index; fractal dimension; continuous, stationary Gaussian process; cosine component; simple linear regression; frequency; bias; variance; asymptotic distribution; covariance; aliasing

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