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Pathway model, superstatistics, Tsallis statistics, and a generalized measure of entropy

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  • Mathai, A.M.
  • Haubold, H.J.

Abstract

The pathway model of Mathai [A pathway to matrix-variate gamma and normal densities, Linear Algebra Appl. 396 (2005) 317–328] is shown to be inferable from the maximization of a certain generalized entropy measure. This entropy is a variant of the generalized entropy of order α, considered in Mathai and Rathie [Basic Concepts in Information Theory and Statistics: Axiomatic Foundations and Applications, Wiley Halsted, New York and Wiley Eastern, New Delhi, 1975], and it is also associated with Shannon, Boltzmann–Gibbs, Rényi, Tsallis, and Havrda–Charvát entropies. The generalized entropy measure introduced here is also shown to have interesting statistical properties and it can be given probabilistic interpretations in terms of inaccuracy measure, expected value, and information content in a scheme. Particular cases of the pathway model are shown to be Tsallis statistics [C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52 (1988) 479–487] and superstatistics introduced by Beck and Cohen [Superstatistics, Physica A 322 (2003) 267–275]. The pathway model's connection to fractional calculus is illustrated by considering a fractional reaction equation.

Suggested Citation

  • Mathai, A.M. & Haubold, H.J., 2007. "Pathway model, superstatistics, Tsallis statistics, and a generalized measure of entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 375(1), pages 110-122.
    • Handle: RePEc:eee:phsmap:v:375:y:2007:i:1:p:110-122
    DOI: 10.1016/j.physa.2006.09.002
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    References listed on IDEAS

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    1. Beck, Christian, 2006. "Stretched exponentials from superstatistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 365(1), pages 96-101.
    2. Saxena, R.K. & Mathai, A.M. & Haubold, H.J., 2004. "On generalized fractional kinetic equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(3), pages 657-664.
    3. Saxena, R.K. & Mathai, A.M. & Haubold, H.J., 2004. "Astrophysical thermonuclear functions for Boltzmann–Gibbs statistics and Tsallis statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(3), pages 649-656.
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    1. Mathai, A.M. & Haubold, H.J., 2008. "Pathway parameter and thermonuclear functions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(11), pages 2462-2470.
    2. Lubashevsky, Ihor & Friedrich, Rudolf & Heuer, Andreas & Ushakov, Andrey, 2009. "Generalized superstatistics of nonequilibrium Markovian systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(21), pages 4535-4550.
    3. Arak M. Mathai & Hans J. Haubold, 2020. "Mathematical Aspects of Krätzel Integral and Krätzel Transform," Mathematics, MDPI, vol. 8(4), pages 1-18, April.
    4. Zhang, Yongping & Shang, Pengjian & Xiong, Hui, 2019. "Multivariate generalized information entropy of financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 1212-1223.
    5. Dhannya Joseph, 2011. "Gamma distribution and extensions by using pathway idea," Statistical Papers, Springer, vol. 52(2), pages 309-325, May.
    6. Sánchez C., Ewin & Vega-Jorquera, Pedro, 2019. "Modelling temporal decay of aftershocks by a solution of the fractional reactive equation," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 43-49.
    7. Jose, K.K. & Naik, Shanoja R., 2008. "A class of asymmetric pathway distributions and an entropy interpretation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(28), pages 6943-6951.
    8. Xu, Meng & Shang, Pengjian & Zhang, Sheng, 2021. "Multiscale Rényi cumulative residual distribution entropy: Reliability analysis of financial time series," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    9. Han, Jung Hun, 2013. "Gamma function to Beck–Cohen superstatistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(19), pages 4288-4298.
    10. Aswathy S. Krishnan & S. M. Sunoj & P. G. Sankaran, 2019. "Quantile-based reliability aspects of cumulative Tsallis entropy in past lifetime," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(1), pages 17-38, January.
    11. Aditya Mani Mishra & Dumitru Baleanu & Fairouz Tchier & Sunil Dutt Purohit, 2019. "Certain Results Comprising the Weighted Chebyshev Function Using Pathway Fractional Integrals," Mathematics, MDPI, vol. 7(10), pages 1-9, September.
    12. Mohammed Z. Alqarni & Mohamed Abdalla, 2023. "Novel Kinds of Fractional λ –Kinetic Equations Involving the Generalized Degenerate Hypergeometric Functions and Their Solutions Using the Pathway-Type Integral," Mathematics, MDPI, vol. 11(19), pages 1-14, October.
    13. Sánchez C., Ewin & Vega-Jorquera, Pedro, 2018. "New Bayesian frequency–magnitude distribution model for earthquakes applied in Chile," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 508(C), pages 305-312.

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