Uniform Minimum Variance Unbiased Estimator of Fractal Dimension





fractal dimension, fractal distribution, uniform minimum variance unbiased estimator


The paper introduced the concept of a fractal distribution using a power-law distribution. It proceeds to determining the relationship between fractal and exponential distribution using a logarithmic transformation of a fractal random variable which turns out to be exponentially distributed. It also considered finding the point estimator of fractional dimension and its statistical characteristics. It was shown that the maximum likelihood estimator of the fractional dimension λ is biased. Another estimator was found and shown to be a uniformly minimum variance unbiased estimator (UMVUE) by Lehmann-Scheffe’s theorem.

Author Biography

Elmer C. Castillano, University of Science and Technology in Southern Philippines, Cagayan de Oro City, Philippines

He is the chairperson, USTP  Math Department


Bauke, H. (2007). Parameter estimation for power-law distributions by maximum likelihood methods. European Physical Journal B, 58,167–173. https://doi.org/10.1140/epjb/e2007-00219-y

Clauset, A., Shalizi, C. R., & Newman, M. E. J. (2009). Power-law distributions in empirical data. SIAM Review, 51(4), 661–703. https://doi.org/ 10.1137/070710111

Feller, W. (1991). An introduction to probability theory and its applications (2nd ed.). New York.

Lehman, P. (1998) Theory of point estimation (2nd ed.). Springer-Verlag.

Malik, H. J. (1970). Estimation of the parameters of the PARETO distribution. Metrika. 15,126–132. https://doi.org/10.1007/BF02613565

Mandelbrot, B. B. (1983). The fractal geometry of nature. W. H. Freeman.

Mandelbrot, B.B., & Van Ness, J.W. (1968). Fractional brownian motions, fractional noises and applications. SIAM Review, 10(4), 422–437 https://www.jstor.org/stable/2027184?seq=1

Newman, M. (2005). Power laws, Pareto distributions and Zipf's law. Contemporary Physics, 46(5), 323-351. https://doi.org/10.1080/00107510500052444

Padua, R.N., Baldado, M., Adanza, J. G.,& Panduyos, J. B. (2013), Statistical analysis of fractal observations: Applications in education and in poverty estimation. SDSSU Multidisciplinary Research Journal, 1(1). https://www.smrj.sdssu.edu.ph/index.php/SMRJ/article/view/84

Padua, R. N.,& Borres, M.S. (2013). From fractal geometry to fractal statistics. Recoletos Multidisciplinary Journal of Research, 1(1). https://doi.org/10.32871/rmrj1301.01.09

Padua, R. N., Ontoy, D. S., Palompon, D. R., & Mirasol, J. M. (2015). Statistical fractal inference. University of the Visayas - Journal of Research, 9(1), 15-22. https://doi.org/10.5281/zenodo.Z144190

Ramachandran, K. M., & Tsokos, C. P. (2009). Mathematical statistics with applications. Elsevier. https://books.google.com.ph/books?id=YFyhXk-ONWwC&printsec=frontcover#v=onepage&q&f=false

Regalado, D. Y., Padua, R.N., Azura, R.B., & Perez, K. B. (2018). Minimum variance unbiased estimation of the scale parameter of exponential distributions and related logarithmic integrals. Journal of Higher Education Research Disciplines, 3(1). https://nmsc.edu.ph/ojs/index.php/jherd/article/view/97

Ross, S.M. (2009). Introduction to probability and statistics for engineers and scientists (4th ed.). Academic Press. https://doi.org/10.1016/B978-0-12-370483-2.X0001-X

Temme, N. M. (2010), Exponential, logarithmic, sine, and cosine integrals. In F.W. Olver, D.W. Lozier, R. F. Boisvert, & C. W. Clark (Eds.), The NIST handbook of mathematical functions. Cambridge University Press.




How to Cite

Maureal, Z. L., Castillano, E. C., & Padua, R. N. (2021). Uniform Minimum Variance Unbiased Estimator of Fractal Dimension. Recoletos Multidisciplinary Research Journal, 9(1), 63–68. https://doi.org/10.32871/rmrj2109.01.06




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