IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v384y2007i2p758-774.html
   My bibliography  Save this article

On measure-theoretic aspects of nonextensive entropy functionals and corresponding maximum entropy prescriptions

Author

Listed:
  • Dukkipati, Ambedkar
  • Bhatnagar, Shalabh
  • Murty, M. Narasimha

Abstract

Shannon entropy of a probability measure P, defined as -∫X(dP/dμ)ln(dP/dμ)dμ on a measure space (X,M,μ), is not a natural extension from the discrete case. However, maximum entropy (ME) prescriptions of Shannon entropy functional in the measure-theoretic case are consistent with those for the discrete case. Also it is well known that Kullback–Leibler relative entropy can be extended naturally to measure-theoretic case. In this paper, we study the measure-theoretic aspects of nonextensive (Tsallis) entropy functionals and discuss the ME prescriptions. We present two results in this regard: (i) we prove that, as in the case of classical relative-entropy, the measure-theoretic definition of Tsallis relative-entropy is a natural extension of its discrete case, and (ii) we show that ME-prescriptions of measure-theoretic Tsallis entropy are consistent with the discrete case with respect to a particular instance of ME.

Suggested Citation

  • Dukkipati, Ambedkar & Bhatnagar, Shalabh & Murty, M. Narasimha, 2007. "On measure-theoretic aspects of nonextensive entropy functionals and corresponding maximum entropy prescriptions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 384(2), pages 758-774.
    • Handle: RePEc:eee:phsmap:v:384:y:2007:i:2:p:758-774
    DOI: 10.1016/j.physa.2007.05.020
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437107005171
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2007.05.020?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Zellner, Arnold & Highfield, Richard A., 1988. "Calculation of maximum entropy distributions and approximation of marginalposterior distributions," Journal of Econometrics, Elsevier, vol. 37(2), pages 195-209, February.
    2. Ryu, Hang K., 1993. "Maximum entropy estimation of density and regression functions," Journal of Econometrics, Elsevier, vol. 56(3), pages 397-440, April.
    3. Ferri, G.L. & Martínez, S. & Plastino, A., 2005. "The role of constraints in Tsallis' nonextensive treatment revisited," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 347(C), pages 205-220.
    4. Anastasiadis, Aristoklis D. & Magoulas, George D., 2004. "Nonextensive statistical mechanics for hybrid learning of neural networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(3), pages 372-382.
    5. Dukkipati, Ambedkar & Murty, M. Narasimha & Bhatnagar, Shalabh, 2006. "Nonextensive triangle equality and other properties of Tsallis relative-entropy minimization," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 361(1), pages 124-138.
    6. Tsallis, Constantino & Stariolo, Daniel A., 1996. "Generalized simulated annealing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 233(1), pages 395-406.
    7. Tsallis, Constantino & Mendes, RenioS. & Plastino, A.R., 1998. "The role of constraints within generalized nonextensive statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 261(3), pages 534-554.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Dukkipati, Ambedkar & Murty, M. Narasimha & Bhatnagar, Shalabh, 2006. "Nonextensive triangle equality and other properties of Tsallis relative-entropy minimization," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 361(1), pages 124-138.
    2. Heckelei, Thomas & Mittelhammer, Ron C., 2003. "Bayesian bootstrap multivariate regression," Journal of Econometrics, Elsevier, vol. 112(2), pages 241-264, February.
    3. Ryu, Hang K. & Slottje, Daniel J., 2000. "Estimating the density of unemployment duration based on contaminated samples or small samples," Journal of Econometrics, Elsevier, vol. 95(1), pages 131-156, March.
    4. Aristoklis D. Anastasiadis & Marcelo P. Albuquerque & Marcio P. Albuquerque & Diogo B. Mussi, 2010. "Tsallis q-exponential describes the distribution of scientific citations—a new characterization of the impact," Scientometrics, Springer;Akadémiai Kiadó, vol. 83(1), pages 205-218, April.
    5. Park, Sung Y. & Bera, Anil K., 2009. "Maximum entropy autoregressive conditional heteroskedasticity model," Journal of Econometrics, Elsevier, vol. 150(2), pages 219-230, June.
    6. Ryu, Hang K. & Slottje, Daniel J., 1996. "Two flexible functional form approaches for approximating the Lorenz curve," Journal of Econometrics, Elsevier, vol. 72(1-2), pages 251-274.
    7. Michael McAleer & Hang K. Ryu & Daniel J. Slottje, 2019. "A New Inequality Measure that is Sensitive to Extreme Values and Asymmetries," Advances in Decision Sciences, Asia University, Taiwan, vol. 23(1), pages 31-61, March.
    8. Rastegin, Alexey E., 2013. "Formulation of the Hellmann–Feynman theorem for the “second choice” version of Tsallis’ thermostatistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(1), pages 103-110.
    9. Domenico Di Gangi & Fabrizio Lillo & Davide Pirino, 2015. "Assessing systemic risk due to fire sales spillover through maximum entropy network reconstruction," Papers 1509.00607, arXiv.org, revised Jul 2018.
    10. Miller, Douglas J. & Liu, Wei-han, 2002. "On the recovery of joint distributions from limited information," Journal of Econometrics, Elsevier, vol. 107(1-2), pages 259-274, March.
    11. Ryu, Hang K. & Slottje, Daniel J., 2017. "Maximum entropy estimation of income distributions from Basmann’s weighted geometric mean measure," Journal of Econometrics, Elsevier, vol. 199(2), pages 221-231.
    12. Chu, Ba, 2011. "Recovering copulas from limited information and an application to asset allocation," Journal of Banking & Finance, Elsevier, vol. 35(7), pages 1824-1842, July.
    13. Naif Alotaibi & A. S. Al-Moisheer & Ibrahim Elbatal & Mansour Shrahili & Mohammed Elgarhy & Ehab M. Almetwally, 2023. "Half Logistic Inverted Nadarajah–Haghighi Distribution under Ranked Set Sampling with Applications," Mathematics, MDPI, vol. 11(7), pages 1, April.
    14. Rubenthaler, Sylvain & Rydén, Tobias & Wiktorsson, Magnus, 2009. "Fast simulated annealing in with an application to maximum likelihood estimation in state-space models," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1912-1931, June.
    15. Moriguchi, Kai & Ueki, Tatsuhito & Saito, Masashi, 2020. "Establishing optimal forest harvesting regulation with continuous approximation," Operations Research Perspectives, Elsevier, vol. 7(C).
    16. Carol Alexander & Jose Maria Sarabia, 2010. "Endogenizing Model Risk to Quantile Estimates," ICMA Centre Discussion Papers in Finance icma-dp2010-07, Henley Business School, University of Reading.
    17. Deeb, Omar El, 2023. "Entropic spatial auto-correlation of voter uncertainty and voter transitions in parliamentary elections," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 617(C).
    18. Ou, Congjie & Huang, Zhifu & Chen, Jincan & El Kaabouchi, A. & Nivanen, L. & Le Méhauté, A. & Wang, Qiuping A., 2009. "A basic problem in the correlations between statistics and thermodynamics," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2313-2318.
    19. Umpierrez, Haridas & Davis, Sergio, 2021. "Fluctuation theorems in q-canonical ensembles," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 563(C).
    20. Zellner, Arnold & Tobias, Justin & Ryu, Hang, 1998. "Bayesian Method of Moments (BMOM) Analysis of Parametric and Semiparametric Regression Models," CUDARE Working Papers 198660, University of California, Berkeley, Department of Agricultural and Resource Economics.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:384:y:2007:i:2:p:758-774. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.