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The Burr XII Distribution Family and the Maximum Entropy Principle: Power-Law Phenomena Are Not Necessarily “Nonextensive”

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DOI: 10.4236/ojs.2015.57073    5,230 Downloads   5,957 Views   Citations
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ABSTRACT

In this paper, we recall for physicists how it is possible using the principle of maximization of the Boltzmann-Shannon entropy to derive the Burr-Singh-Maddala (BurrXII) double power law probability distribution function and its approximations (Pareto, loglogistic.) and extension (GB2…) first used in econometrics. This is possible using a deformation of the power function, as this has been done in complex systems for the exponential function. We give to that distribution a deep stochastic interpretation using the theory of Weron et al. Applied to thermodynamics, the entropy nonextensivity can be accounted for by assuming that the asymptotic exponents are scale dependent. Therefore functions which describe phenomena presenting power-law asymptotic behaviour can be obtained without introducing exotic forms of the entropy.

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The authors declare no conflicts of interest.

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Brouers, F. (2015) The Burr XII Distribution Family and the Maximum Entropy Principle: Power-Law Phenomena Are Not Necessarily “Nonextensive”. Open Journal of Statistics, 5, 730-741. doi: 10.4236/ojs.2015.57073.

References

[1] Burr, I.W. (1942) Cumulative Frequency Functions. The Annals of Mathematical Statistics, 13, 215-232.
http://dx.doi.org/10.1007/s10463-011-0342-9
[2] Singh, S. and Maddala, G. (1976) A Function for the Size Distribution of Income. Econometrica, 44, 963-970.
http://dx.doi.org/10.1007/s10985-005-2970-y
[3] Park, S.Y. and Bera, A.K. (2007) Maximum Entropy Income Densities with an Application to the U.S. Personal Income Data. Working Paper, University of Illinois, Urbana-Champaign.
http://dx.doi.org/10.1007/s11634-008-0025-4
[4] Park, S.Y. and Bera, A.K. (2009) Maximum Entropy Autoregressive Conditional Heteroskedasticity Model. Journal of Econometrics, 150, 219-230.
http://dx.doi.org/10.1007/s10463-009-0239-z
[5] Papalexiou, S.M. and Koutsoyiannis, D. (2012) Entropy Based Derivation of Probability Distributions: A Case Study to Daily Rainfall. Advances in Water Resources, 45, 51-57.
http://dx.doi.org/10.1007/s12561-009-9000-7
[6] McDonald, J.B. (1984) Some Generalized Functions for the Size Distribution of Income. Econometrica: Journal of the Econometric Society, 52, 647-663.
http://dx.doi.org/10.1007/s10255-011-0093-7
[7] Ferriani, F. (2015) Traders and Time: Who Moves the Market? Studies in Economics and Finance, 32, 74-97.
http://dx.doi.org/10.1016/j.jmva.2005.03.002
[8] Girard, S. and Guillou, A. (2015) Reduced-Bias Estimator of the Conditional Tail Expectation of Heavy-Tailed Distributions. In: Editor, Ed., Mathematical Statistics and Limit Theorems, Springer International Publishing, 105-123.
http://dx.doi.org/10.3150/bj/1137421639
[9] Streftaris, G., Waters, H.R. and Stott, A.D. (2015) The Effect of Model Uncertainty on the Pricing of Critical Illness Insurance. Annals of Actuarial Science, 9, 108-133.
[10] Ducey, M.J. and Gove, J.H. (2015) Size-Biased Distributions in the Generalized Beta Distribution Family, with Applications to Forestry. Forestry, 88, 143-151.
http://dx.doi.org/10.1006/jmva.2001.2012
[11] Brouers, F. (2014) Statistical Foundation of Empirical Isotherms. Open Journal of Statistics, 4, 687-701.
http://dx.doi.org/10.4236/ojs.2014.49064
[12] Brouers, F. and Al-Musawi, T.J. (2015) On the Optimum Use of Isotherm Model for the Characterization of Biosorption of Lead onto Algae. Journal of Molecular Liquids, 212, 46-51.
http://dx.doi.org/10.1016/j.molliq.2015.08.054
[13] Brouers, F. and Sotolongo-Costa, O. (2006) Generalized Fractal Kinetics in Complex Systems (Application to Biophysics and Biotechnology). Physica A: Statistical Mechanics and Its Applications, 368, 165-175.
http://dx.doi.org/10.1016/j.physa.2005.12.062
[14] Hamissa, A.M.B., Brouers, F., Mahjoub, B. and Seffen, M. (2007) Adsorption of Textile Dyes Using Agave americana (L.) Fibres: Equilibrium and Kinetics Modelling. Adsorption Science and Technology, 25, 311-325.
http://dx.doi.org/10.1260/026361707783432533
[15] Ncibi, M.C., Mahjoub, B., Seffen, M., Brouers, F. and Gaspard, S. (2009) Sorption Dynamic Investigation of Chromium (VI) onto Posidonia oceanic Fibres: Kinetic Modelling Using New Generalized Fractal Equation. Biochemical Engineering Journal, 46, 141-146.
http://dx.doi.org/10.1016/j.bej.2009.04.022
[16] Figaro, S., Avril, J.P., Brouers, F., Ouensanga, A. and Gaspard, S. (2009) Adsorption Studies of Molasse’s Wastewaters on Activated Carbon: Modelling with a New Fractal Kinetic Equation and Evaluation of Kinetic Models. Journal of Hazardous Materials, 161, 649-656.
http://dx.doi.org/10.1016/j.jhazmat.2008.04.006
[17] Hamissa, A.B., Brouers, F., Ncibi, M.C. and Seffen, M. (2013) Kinetic Modeling Study on Methylene Blue Sorption onto Agave americana Fibers: Fractal Kinetics and Regeneration Studies. Separation Science and Technology, 48, 2834-2842.
http://dx.doi.org/10.1080/01496395.2013.809104
[18] Debord, J., Bollinger, J.C., Bordas, F., Harel, M. and Dantoine, T. (2013) Microcalorimetric Study of the Inhibition of Butyrylcholinesterase by Sodium Arsenite and Zinc Chloride: Use of a Fractal Kinetic Model. Thermochimica Acta, 561, 49-53.
http://dx.doi.org/10.1016/j.tca.2013.03.027
[19] Kyzas, G.Z., Deliyanni, E.A. and Lazaridis, N.K. (2014) Magnetic Modification of Microporous Carbon for Dye Adsorption. Journal of Colloid and Interface Science, 430, 166-173.
http://dx.doi.org/10.1016/j.jcis.2014.05.049
[20] Brouers, F. (2014) The Fractal (BSf) Kinetics Equation and Its Approximations. Journal of Modern Physics, 5, 1594-1598.
http://dx.doi.org/10.4236/jmp.2014.516160
[21] Pereira, L.M. (2010) Fractal Pharmacokinetics. Computational and Mathematical Methods in Medicine, 11, 161-184.
http://dx.doi.org/10.1080/17486700903029280
[22] Weron, K. and Kotulski, M. (1997) On the Equivalence of the Parallel Channel and the Correlated Cluster Relaxation models. Journal of Statistical Physics, 88, 1241-1256.
http://dx.doi.org/10.1007/BF02732433
[23] Brouers, F. and Sotolongo-Costa, O. (2005) Relaxation in Heterogeneous Systems: A Rare Events Dominated Phenomenon. Physica A: Statistical Mechanics and Its Applications, 356, 359-374.
http://dx.doi.org/10.1016/j.physa.2005.03.029
[24] Abubakar, M., Aliyu, A.B. and Ahmad, N. (2015) Flexural Strength Analysis of Dense and Porous Sintered Clay Using Weibull Probability Distribution. Applied Mechanics and Materials, 761, 347-351.
http://dx.doi.org/10.4028/www.scientific.net/AMM.761.347
[25] Tu, J.W., Guo, D.L., Mei, S.T., Jiang, H.C. and Li, X.P. (2015) Three-Parameter Weibull Distribution Model for Tensile Strength of GFRP Bars Based on Experimental Tests. Materials Research Innovations, 19, S5-1191-S5-1196.
http://dx.doi.org/10.1179/1432891714z.0000000001276
[26] Bütikofer, L., Stawarczyk, B. and Roos, M. (2015) Two Regression Methods for Estimation of a Two-Parameter Weibull Distribution for Reliability of Dental Materials. Dental Materials, 31, e33-e50.
http://dx.doi.org/10.1016/j.dental.2014.11.014
[27] Nadler, D.L. (2015) Developing a Weibull Model Extension to Estimate Cancer Latency Times. Doctoral Dissertation, State University of New York at Albany, Albany.
[28] Goutelle, S., Maurin, M., Rougier, F., Barbaut, X., Bourguignon, L., Ducher, M. and Maire, P. (2008) The Hill Equation: A Review of Its Capabilities in Pharmacological Modelling. Fundamental and Clinical Pharmacology, 22, 633-648.
http://dx.doi.org/10.1111/j.1472-8206.2008.00633.x
[29] Hill Keller, F., Schröppel, B. and Ludwig, U. (2015) Pharmacokinetic and Pharmacodynamics Considerations of Antimicrobial Drug Therapy in Cancer Patients with Kidney Dysfunction. World Journal of Nephrology, 4, 330-344.
http://dx.doi.org/10.5527/wjn.v4.i3.330
[30] Chou, T.C. (1976) Derivation and Properties of Michaelis-Menten Type and Hill Type Equations for Reference Ligands. Journal of Theoretical Biology, 59, 253-276.
http://dx.doi.org/10.1016/0022-5193(76)90169-7
[31] Bhaskaran, S., Umesh, P. and Nair, A.S. (2015) Hill Equation in Modeling Transcriptional Regulation. In: Singh, V. and Dhar, P.K., Eds., Systems and Synthetic Biology, Springer, Dordrecht, 77-92.
http://dx.doi.org/10.1007/978-94-017-9514-2_5
[32] Criado, M.N., Civera, M., Martínez, A. and Rodrigo, D. (2015) Use of Weibull Distribution to Quantify the Antioxidant Effect of Stevia rebaudiana on Oxidative Enzymes. LWT-Food Science and Technology, 60, 985-989.
http://dx.doi.org/10.1016/j.lwt.2014.10.041
[33] Oral, F., Ekmekçi, I. and Onat, N. (2015) Weibull Distribution for Determination of Wind Analysis and Energy Production. World Journal of Engineering, 12, 215-220.
http://dx.doi.org/10.1260/1708-5284.12.3.215
[34] Brouers, F., Sotolongo, O., Marquez, F. and Pirard, J.P. (2005) Microporous and Heterogeneous Surface Adsorption Isotherms Arising from Levy Distributions. Physica A: Statistical Mechanics and Its Applications, 349, 271-282.
http://dx.doi.org/10.1016/j.physa.2004.10.032
[35] Ncibi, M.C., Altenor, S., Seffen, M., Brouers, F. and Gaspard, S. (2008) Modelling Single Compound Adsorption onto Porous and Non-Porous Sorbents Using a Deformed Weibull Exponential Isotherm. Chemical Engineering Journal, 145, 196-202.
http://dx.doi.org/10.1016/j.cej.2008.04.001
[36] Yuhn, K.H., Kim, S.B. and Nam, J.H. (2015) Bubbles and the Weibull Distribution: Was There an Explosive Bubble in US Stock Prices before the Global Economic Crisis? Applied Economics, 47, 255-271.
http://dx.doi.org/10.1080/00036846.2014.969824
[37] Sornette, D. (2003) Critical Phenomena in Natural Sciences. 2nd Edition, Chap. 14, Springer, Heidelberg.
[38] Reinyi, A. (1970) Probability Theory. North-Holland Publishing Company, Amsterdam.
[39] Tsallis, C. (1988) Possible Generalization of Boltzmann-Gibbs Statistics. Journal of Statistical Physics, 52, 479-487.
http://dx.doi.org/10.1007/BF01016429
[40] Tsallis, C. (2009) Nonadditive Entropy and Nonextensive Statistical Mechanics—An Overview after 20 Years. Brazilian Journal of Physics, 39, 337-356.
http://dx.doi.org/10.1590/S0103-97332009000400002
[41] Kapur, J. and Kesavan, H. (1992) Entropy Optimization Principles with Applications. Academic Press, Waltham.
http://dx.doi.org/10.1007/978-94-011-2430-0_1
[42] Wu, X. (2003) Calculation of Maximum Entropy Densities with Application to Income Distribution. Journal of Econometrics, 115, 347-354.
http://dx.doi.org/10.1016/S0304-4076(03)00114-3
[43] Rathie, P.N. and Da Silva, S. (2008) Shannon, Lévy, and Tsallis: A Note. Applied Mathematical Sciences, 2, 1359-1363.
[44] Kafri, O. (2009) The Distributions in Nature and Entropy Principle. arXiv:0907.4852.
[45] Papalexiou, S.M. and Koutsoyiannis, D. (2011) Entropy Maximization, p-Moments and Power-Type Distributions in Nature. European Geosciences Union General Assembly 2011, Geophysical Research Abstracts, Vol. 13, Vienna, 03-08 April 2011, European Geosciences Union, EGU2011-6884.
[46] Mahmoud, M.R. and El Ghafour, A.S.A. (2013) Shannon Entropy for the Generalized Feller-Pareto (GFP) Family and Order Statistics of GFP Subfamilies. Applied Mathematical Sciences, 7, 3247-3253.
[47] Peterson, J., Dixit, P.D. and Dill, K.A. (2013) A Maximum Entropy Framework for Nonexponential Distributions. Proceedings of the National Academy of Sciences of the United States of America, 110, 20380-20385.
http://dx.doi.org/10.1073/pnas.1320578110
[48] Visser, M. (2013) Zipf’s Law, Power Laws and Maximum Entropy. New Journal of Physics, 15, Article ID: 043021.
http://dx.doi.org/10.1088/1367-2630/15/4/043021
[49] Freundlich, U. (1906) Uber Die Adsorption in Losungen. Journal of physical chemistry, 57, 385-470.
[50] Kopelman, R. (1988) Fractal Reaction Kinetics. Science, 241, 1620-1626.
http://dx.doi.org/10.1126/science.241.4873.1620
[51] Savageau, M.A. (1995) Michaelis-Menten Mechanism Reconsidered: Implications of Fractal Kinetics. Journal of Theoretical Biology, 176, 115-124.
http://dx.doi.org/10.1006/jtbi.1995.0181
[52] van den Broek, J. and Nishiura, H. (2009) Using Epidemic Prevalence Data to Jointly Estimate Reproduction and Removal. The Annals of Appliedstatistics, 3, 1505-1520.
http://dx.doi.org/10.1214/09-aoas270
[53] Verhulst, P.F. (1845) Nouveaux Mémoires de l’Académie des Sciences, des Arts et des Beaux Arts de Belgique. Vol. 18.
[54] Ausloos, M. and Dirickx, M. (Eds.) (2006) The Logistic Map and the Route to Chaos: From the Beginnings to Modern Applications. Springer Science and Business Media, Berlin.
http://dx.doi.org/10.1007/3-540-32023-7
[55] Havriliak, S. and Negami, S. (1967) A Complex Plane Representation of Dielectric and Mechanical Relaxation Processes in Some Polymers. Polymer, 8, 161-210.
http://dx.doi.org/10.1016/0032-3861(67)90021-3
[56] Langmuir, I. (1918) The Adsorption of Gases on Plane Surfaces of Glass, Mica and Platinum. Journal of the American Chemical Society, 40, 1361-1403.
http://dx.doi.org/10.1021/ja02242a004
[57] Sips, R. (1948) The Structure of a Catalyst Surface. The Journal of Chemical Physics, 16, 490-495.
http://dx.doi.org/10.1063/1.1746922
[58] Jurlewicz, A. and Weron, K. (1999) A General Probabilistic Approach to the Universal Relaxation Response of Complex Systems. Cellular and Molecular Biology Letters, 4, 55-86.
[59] Stanislavsky, A. and Weron, K. (2013) Is There a Motivation for a Universal Behaviour in Molecular Populations Undergoing Chemical Reactions? Physical Chemistry Chemical Physics, 15, 15595-15601.
http://dx.doi.org/10.1039/c3cp52272e
[60] Rodriguez, R.N. (1977) A Guide to the Burr Type XII Distributions. Biometrika, 64, 129-134.
http://dx.doi.org/10.1093/biomet/64.1.129
[61] Brouers, F. (2013) Sorption Isotherms and Probability Theory of Complex Systems. arXiv:1309.5340.
[62] Kapur, J.N. (1989) Maximum-Entropy Models in Science and Engineering. John Wiley and Sons, New York.
[63] Beck, C. (2002) Non-Additivity of Tsallis Entropies and Fluctuations of Temperature. Europhysics Letters (EPL), 57, 329-333.
http://dx.doi.org/10.1209/epl/i2002-00464-8
[64] Brouers, F., Sotolongo-Costa, O. and Weron, K. (2004) Burr, Lévy, Tsallis. Physica A: Statistical Mechanics and Its Applications, 344, 409-416.
http://dx.doi.org/10.1016/j.physa.2004.06.008
[65] Brouers, F. and Sotolongo-Costa, O. (2003) Universal Relaxation in Nonextensive Systems. Europhysics Letters (EPL), 62, 808-814.
http://dx.doi.org/10.1209/epl/i2003-00444-0
[66] Brouers, F., Sotolongo-Costa, O., Gonzalez, A. and Pirard, J.P. (2005) Entropic Origin of Dielectric Relaxation Universalities in Heterogeneous Materials (Polymers, Glasses, Aerogel Catalysts). Physica Status Solidi (C), 2, 3529-3531.
http://dx.doi.org/10.1002/pssc.200461736

  
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