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Biometrika 1993 80(1):246-251; doi:10.1093/biomet/80.1.246
© 1993 by Biometrika Trust
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MISCELLANEA

On the performance of box-counting estimators of fractal dimension

PETER HALL and ANDREW WOOD

Centre for Mathematics and its Applications, Australian National University GPO Box 4, Canberra, A.C.T. 2601, Australia

Box-counting estimators are popular for estimating fractal dimension. However, very little is known of their stochastic properties, despite increasing statistical interest in their application. We show that, if the irregular curve to which the estimators are applied is modelled by a Gaussian process, concise formulae may be developed for asymptotic bias and variance of box-counting estimators. These formulae point to critical differences between a native form of the box-counting estimator, based directly on the capacity definition of fractal dimension, and a regression-inspired version of that estimator.

Key Words: Bias • Box-counting • Convergence rate • Fractal dimension • Gaussian process • Hausdorff dimension • Regression


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