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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., {"references":["Devaney, R. L. (1988). Fractal patterns arising in chaotic dynamical systems. In H. O. Peitgen & D Saupe (Eds.). The Science of Fractal Images (pp. 137-168). New York, USA: New York.","Fay, M. P., & Proschan, M. A. (2010). WilcoxonMann-Whitney or t-test? On assumptions for hypothesis tests and multiple interpretations of decision rules. Statistics Surveys, 4, 1-39.","Gargantini, A., & Fraser, G. (2011). Generating minimal fault detecting test suites for general Boolean specifi cations. Information and Software Technology, 53(11), 12631273.","Huber, P. J. (2011). Robust statistics. In M. Lovric (Ed.) International encyclopedia of statistical science (pp. 1248-1251). Switzerland: Springer Berlin Heidelberg.","Mandelbrot, B. B. (1967). How long is the coast of Britain. Science, 156(3775), 636-638.","Mann, H. B., & Whitney, D. R. (1947). On a test of whether one of two random variables is stochastically larger than the other. The Annals of Mathematical Statistics, 8(1), 5060.","McKnight, P. E., & Najab, J. (2010). Mann-Whitney U Test. Corsini Encyclopedia of Psychology. New Jersey, USA: John Wiley & Sons, Inc.","Motulsky, H. J. (2007). Prism 5 statistics guide, 2007. GraphPad Software, 31(1), 39-42.","Nachar, N. (2008). The Mann-Whitney U: A test for assessing whether two independent samples come from the same distribution. Tutorials in Quantitative Methods for Psychology, 4(1), 13-20.","Padua, R. N., Barabat, E. O., Borres, M. S., & Salazar, R. K. (2013). Some Results on Multifractal Spectral Analysis. Recoletos Multidisciplinary Research Journal, 1(2), 57-63.","Selvam, A . M. (2009). Fractal fluctuations and statistical normal distribution. Fractals, 17(03), 333349"]}

Multifractal probability distributions are defined as mixture of n monofractal distributions. The exponents are the fractal dimensions which determine the information- filling property of an observation from this distribution. A useful device for examining multifractal observations is the multifractal spectrum. The fractal spectrum was shown to be a one–to–one function, monotonically increasing with x on a logarithmic scale for non–fractal distributions. Similarly, multifractal distributions viewed on a larger scale (s) have a spectrum that behaves like power function but when viewed on a smaller scale, it behaves like a concave downward quadratic function where A, B and C are parameters to be estimated., {"references":["Lapenna, V., Macchiato, M., & Telesca, L. (1998). 1/f^β Fluctuations and self-similarity in earthquakes dynamics: observational evidences in southern Italy. Physics of the Earth and Planetary Interiors, 106(1), 115-127.","Padua, R. N., Barabat, E. O., Borres, M. S., & Salazar, R. K. (2013). Some Results on Multifractal Spectral Analysis. Recoletos Multidisciplinary Research Journal, 1(2), 57-63.","Padua, R., Barabat, E., & Regalado, D. Y. (2013). A simple test for mono and multifractality of statistical observations, NORSU Prism Journal of Higher Education, 23-24.","Panduyos, J. B., Villanueva, F. P., & Padua, R. N. (2014). Fitting a Fractal Distribution on Philippine Seismic Data: 20112013. SDSSU Multidisciplinary Research Journal, 1(1).","Selvam, A . M. (2009). Fractal fl uctuations and statistical normal distribution. Fractals, 17(03), 333-349.","Tukey, J. W. (1972). How computing and statistics affect each other. The Babbage Memorial Meeting: Report of Proceedings. London: British Computer Society"]}