A ML Estimator of the Correlation Dimension for Left-hand Truncated Data Samples

A maximum-likelihood (ML) estimator of the correlation dimension d2 of fractal sets of points not affected by the left-hand truncation of their inter-distances is defined. Such truncation might produce significant biases of the ML estimates of d2 when the observed scale range of the phenomenon is very narrow, as often occurs in seismological studies. A second very simple algorithm based on the determination of the first two moments of the inter-distances distribution (SOM) is also proposed, itself not biased by the left-hand truncation effect. The asymptotic variance of the ML estimates is given. Statistical tests carried out on data samples with different sizes extracted from populations of interdistances following a power law, suggested that the sample variance of the estimates obtained by the proposed methods are not significantly different, and are well estimated by the asymptotic variance also for samples containing a few hundred inter-distances. To examine the effects of different sources of systematic errors, the two estimators were also applied to sets of inter-distances between points belonging to statistical fractal distributions, baker’s maps and experimental distributions of earthquake epicentres. For a full evaluation of the results achieved by the methods proposed here, these were compared with those obtained by the ML estimator for untruncated samples or by the least-squares algorithm.