Statistical precision of dimension estimators

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Abstract
We address the issue of statistical error in the estimation of the correlation dimension of a (possibly fractal) set from a finite sample of N points on the set. Although both numerical and analytical investigations show that this error generically scales with the number of points as O(1/ √N ) for N→∞, it is also shown that this is not always the case. The exceptions exhibit anomalously precise O(1/N) scaling, and can be characterized by their extreme uniformity. The existence of these exceptions points out that the central limit theorem is not directly applicable. The exceptions are not generic, but it is common for a fractal to exhibit 1/N scaling that crosses over to 1/ √N scaling only after N becomes very large. The coefficients of both the 1/ √N term and the 1/N term are derived, and the results are applied to practical problems in dimension estimation. In particular, the practice of computing dimension with only a few reference points is discussed. The choice of optimal scaling region and the overall scaling of error (systematic and statistical) with N are also considered in light of these results.

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