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

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.

Information entropy of the generalized beta distribution

“…For the Pareto family of distributions, entropy formulas are given in the paper [3,6]. In papers [7,8,9] you can find entropy formulas for such subfamilies as Burr type XII, loglogistic and Lomax. Burr's type XII distribution is widely used in technical and physical systems in the analysis of survival as a more flexible alternative than the Weibull distribution [7].…”

Dubinin isotherms versus the Brouers–Sotolongo family isotherms: A case study

“…The knowledge of a, b, and c allows the calculation of the usual statistical quantities of the distribution. It obeys a birth and death differential equation which is discussed in Brouers (2014aBrouers ( , 2014bBrouers ( , 2015. As shown in detail in Brouers (2014b), some of the most popular empirical and semiempirical isotherms can be derived simply from the GBS isotherm, and others are purely empirical such as the Freundlich (1906), the Redlich-Peterson (1959), and the Generalized Freundlich isotherms (Sips, 1950).…”

“…Isotherms have been used for decades in the gaseous or aqueous phase with two main objectives, first to obtain information on the nature of the sorbent surface of porous materials (sorption energy, type of porosity, and type of heterogeneity) in order to use them, taking advantage of their large specific surface, and second, to prepare and characterize specific sorbent to eliminate particular molecules in the treatment of water and more generally for the purpose of physical, chemical, and biological (Burr, 1942;Singh and Maddala, 1976) or q-Weibull. It is a natural extension to natural and physico-chemical systems of the classical concept of exponential and appears naturally in attempts to generalize the classical thermodynamic to complex systems (Brouers, 2013(Brouers, , 2015Tsallis, 1988). In this paper, we want to demonstrate that the Dubinin isotherms belong asymptotically to the same family and analyze the various approximations using data on benzene taken from one of the original Dubinin papers and analyzing in the same manner the data published in the PhD thesis of one of us.…”

Entropy-Based Parameter Estimation for the Four-Parameter Exponential Gamma Distribution

“…For example, an entropy-based derivation of daily rainfall probability distribution [24], the Burrr XII-Singh-Maddala (BSM) distribution function derived from the maximum entropy principle using the Boltzmann-Shannon entropy with some constraints [25]. "Entropy-Based Parameter Estimation in Hydrology" is the first book focusing on parameter estimation using entropy for a number of distributions frequently used in hydrology [3], including the uniform distribution, exponential distribution, normal distribution, two-parameter lognormal distribution, three-parameter lognormal distribution, extreme value type I distribution, log-extreme value type I distribution, extreme value type III distribution, generalized extreme value distribution, Weibull distribution, gamma distribution, Pearson type III distribution, log-Pearson type III distribution, beta distribution, two-parameter log-logistic distribution, three-parameter log-logistic distribution, two-parameter Pareto distribution, two-parameter generalized Pareto distribution, three-parameter generalized Pareto distribution and two-component extreme value distribution.…”