Statistical fractal inference

Abstract

The inverse power law distributions are used as the model for fractal probability distributions that have fractional exponents (λ) and such that the transformation X(1-λ) is uniformly distributed. The paper examines aspects of point estimation and tests of hypotheses about statistical fractals. It is shown that the maximum likelihood estimator of the fractional dimension λ is uniformly minimum variance unbiased estimator (UMVUE) using the Lehmann-Scheffe’s theorem and also that the likelihood ratio test for H0: λ= λ0 is uniformly most powerful (UMPT) by the Neymann-Pearson Lemma. The paper likewise explains that the test for equality of two medians of two fractal distributions is equivalent to a test for the equality of the fractal dimensions. In fact, the result is generalized to the test for the equality of two αth quantiles of two fractal distributions. The test for the equality of two fractal distributions is compared with the Mann-Whitney U test and with the Student’s t-test for independent samples in terms of robustness and asymptotic Pitmann efficiency.