IDEAS home Printed from https://ideas.repec.org/p/zbw/iwqwdp/072015.html
   My bibliography  Save this paper

Cumulative Paired 饾湙-Entropy

Author

Listed:
  • Klein, Ingo
  • Mangold, Benedikt

Abstract

A new kind of entropy will be introduced generalizing both the differential entropy and the cumulative (residual) entropy. The generalization is twofold. Firstly, we define the entropy for cumulative distribution functions (cdf) and survivor functions (sf) simultaneously instead of densities, cdf or sf alone. Secondly, we consider a general 'entropy generating function' 饾湙 like Burbea & Rao (1982) or Liese & Vajda (1987) in the context of 饾湙-divergences. Combining the ideas of a 饾湙-entropy and a cumulative entropy gives the new 'cumulative paired 饾湙-entropy' (CPE饾湙). With some modifications or simplifications this new entropy has already been discussed in at least four scientific disciplines. In the fuzzy set theory cumulative paired 饾湙- entropies were defined for membership functions. A discrete version serves as a measure of dispersion for ordered categorial variables. More recently, uncertainty and reliability theory considered some variants as a measure of information. With only one exception the discussions seem to happen independently of each other. We consider CPE饾湙 only for continuous cdf and show that CPE 饾湙 is rather a measure of dispersion than a measure of information. At first, this will be demonstrated by deriving an upper bound which is determined by the standard deviation and by solving the maximum entropy problem under the restriction that the variance is fixed. We cannot only reproduce the central role of the logistic distribution in entropy maximization. We derive Tukey's Lambda distribution as the solution of an entropy maximization problem as well. Secondly, it will be shown explicitly that CPE饾湙 fulfills the axioms of a dispersion measure. The corresponding dispersion functional can easily be estimated by an L-estimator with all its known asymptotical properties. CPE饾湙 are the starting point for several related concepts like mutual 饾湙-information, 饾湙-correlation and 饾湙-regression which generalize Gini correlation and Gini regression. We give a short introduction into all of these related concepts. Also linear rank tests for scale can be developed based on the new entropy. We show that almost all known tests are special cases and introduce some new tests. In the literature Shannon's differential entropy has been calculated for a lot of distributions. The formulas were presented explicitly. We have done the same for CPE饾湙 if the cdf is available in a closed form.

Suggested Citation

  • Klein, Ingo & Mangold, Benedikt, 2015. "Cumulative Paired 饾湙-Entropy," FAU Discussion Papers in Economics 07/2015, Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics.
    • Handle: RePEc:zbw:iwqwdp:072015
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a search for a similarly titled item that would be available.

    References listed on IDEAS

    as
    1. Shlomo Yitzhaki & Edna Schechtman, 2005. "The properties of the extended Gini measures of variability and inequality," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilit脿 e Statistiche Applicate - University of Rome, vol. 0(3), pages 401-433.
    2. Sunoj, S.M. & Sankaran, P.G., 2012. "Quantile based entropy function," Statistics & Probability Letters, Elsevier, vol. 82(6), pages 1049-1053.
    3. Tamar Gadrich & Emil Bashkansky & Ri膷ardas Zitikis, 2015. "Assessing variation: a unifying approach for all scales of measurement," Quality & Quantity: International Journal of Methodology, Springer, vol. 49(3), pages 1145-1167, May.
    4. B茅n茅dicte Apouey & Jacques Silber, 2013. "Inequality and Bi-Polarization in Socioeconomic Status and Health: Ordinal Approaches," Research on Economic Inequality, in: Health and Inequality, volume 21, pages 77-109, Emerald Group Publishing Limited.
    5. Schechtman, E. & Yitzhaki, S., 1999. "On the proper bounds of the Gini correlation," Economics Letters, Elsevier, vol. 63(2), pages 133-138, May.
    6. Burger, H. U., 1993. "Dispersion orderings with applications to nonparametric tests," Statistics & Probability Letters, Elsevier, vol. 16(1), pages 1-9, January.
    7. Hall, Peter & Wolff, Rodney C. L. & Yao, Qiwei, 1999. "Methods for estimating a conditional distribution function," LSE Research Online Documents on Economics 6631, London School of Economics and Political Science, LSE Library.
    8. Abul Naga, Ramses H. & Yalcin, Tarik, 2008. "Inequality measurement for ordered response health data," Journal of Health Economics, Elsevier, vol. 27(6), pages 1614-1625, December.
    9. V. Zardasht & S. Parsi & M. Mousazadeh, 2015. "On empirical cumulative residual entropy and a goodness-of-fit test for exponentiality," Statistical Papers, Springer, vol. 56(3), pages 677-688, August.
    10. Shlomo Yitzhaki, 2003. "Gini鈥檚 Mean difference: a superior measure of variability for non-normal distributions," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilit脿 e Statistiche Applicate - University of Rome, vol. 0(2), pages 285-316.
    11. Allison, R. Andrew & Foster, James E., 2004. "Measuring health inequality using qualitative data," Journal of Health Economics, Elsevier, vol. 23(3), pages 505-524, May.
    12. Lerman, Robert I. & Yitzhaki, Shlomo, 1984. "A note on the calculation and interpretation of the Gini index," Economics Letters, Elsevier, vol. 15(3-4), pages 363-368.
    13. Buhong Zheng, 2011. "A new approach to measure socioeconomic inequality in health," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 9(4), pages 555-577, December.
    14. Ebrahimi, Nader & Maasoumi, Esfandiar & Soofi, Ehsan S., 1999. "Ordering univariate distributions by entropy and variance," Journal of Econometrics, Elsevier, vol. 90(2), pages 317-336, June.
    15. E. Schechtman & S. Yitzhaki, 2003. "A Family of Correlation Coefficients Based on the Extended Gini Index," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 1(2), pages 129-146, August.
    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. SILBER, Jacques & XU, Yongsheng, 2016. "The Health Equivalent Adjusted Level (HEAL): Taking an Ordinal Approach to the Measurement of a Society's Health Achievements," Discussion paper series HIAS-E-31, Hitotsubashi Institute for Advanced Study, Hitotsubashi University.
    2. Val茅rie B茅renger & Jacques Silber, 2022. "On the Measurement of Happiness and of its Inequality," Journal of Happiness Studies, Springer, vol. 23(3), pages 861-902, March.
    3. Frank A Cowell & Martyna Kobus & Radoslaw Kurek, 2017. "Welfare and Inequality Comparisons for Uni- and Multi-dimensional Distributions of Ordinal Data," STICERD - Public Economics Programme Discussion Papers 31, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    4. Hongliang Wang & Yiwen Yu, 2016. "Increasing health inequality in China: An empirical study with ordinal data," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 14(1), pages 41-61, March.
    5. Fatiha Bennia & Nicolas Gravel & Brice Magdalou & Patrick Moyes, 2022. "Is body weight better distributed among men than among women? A robust normative analysis for France, the UK, and the US," Scandinavian Journal of Economics, Wiley Blackwell, vol. 124(1), pages 69-103, January.
    6. Yoel Finkel & Yevgeny Artsev & Shlomo Yitzhaki, 2006. "Inequality measurement and the time structure of household income in Israel," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 4(2), pages 153-179, August.
    7. Carmen Herrero & Antonio Villar, 2013. "On the Comparison of Group Performance with Categorical Data," PLOS ONE, Public Library of Science, vol. 8(12), pages 1-7, December.
    8. Paul Makdissi & Myra Yazbeck, 2017. "Robust rankings of socioeconomic health inequality using a categorical variable," Health Economics, John Wiley & Sons, Ltd., vol. 26(9), pages 1132-1145, September.
    9. Maria Livia 艦TEF膫NESCU, 2015. "Analyzing the health status of the population using ordinal data," Computational Methods in Social Sciences (CMSS), "Nicolae Titulescu" University of Bucharest, Faculty of Economic Sciences, vol. 3(1), pages 18-24, June.
    10. Nicolas Gravel & Brice Magdalou & Patrick Moyes, 2021. "Ranking distributions of an ordinal variable," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(1), pages 33-80, February.
    11. St茅phane Mussard & Fattouma Souissi-Benrejab, 2019. "Gini-PLS Regressions," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 17(3), pages 477-512, September.
      • St茅phane Mussard & Fattouma Souissi-Benrejab, 2015. "Gini-PLS Regressions," Working Papers 15-03, LAMETA, Universtiy of Montpellier, revised Feb 2015.
      • Stephane Mussard & Fattouma Souissi-Benrejab, 2015. "Gini-PLS Regressions," Cahiers de recherche 17-02, Departement d'脙漏conomique de l'脙鈥癱ole de gestion 脙 l'Universit脙漏 de Sherbrooke, revised Jan 2017.
    12. Suman Seth and Gaston Yalonetzky, 2018. "Assessing Deprivation with Ordinal Variables: Depth Sensitivity and Poverty Aversion," OPHI Working Papers ophiwp123.pdf, Queen Elizabeth House, University of Oxford.
    13. Joan Costa Font & Frank Cowell, 2013. "Measuring Health Inequality with Categorical Data: Some Regional Patterns," Research on Economic Inequality, in: Health and Inequality, volume 21, pages 53-76, Emerald Group Publishing Limited.
    14. Kobus, Martyna & Mi艂o艣, Piotr, 2012. "Inequality decomposition by population subgroups for ordinal data," Journal of Health Economics, Elsevier, vol. 31(1), pages 15-21.
    15. Martyna Kobus & Rados艂aw Kurek, 2019. "Multidimensional polarization for ordinal data," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 17(3), pages 301-317, September.
    16. Shelef, Amit, 2016. "A Gini-based unit root test," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 763-772.
    17. Joan Costa-Font & Frank A. Cowell, 2022. "The measurement of health inequalities: does status matter?," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 20(2), pages 299-325, June.
    18. Marta Pascual & David Cantarero & Paloma Lanza, 2018. "Health polarization and inequalities across Europe: an empirical approach," The European Journal of Health Economics, Springer;Deutsche Gesellschaft f眉r Gesundheits枚konomie (DGG脰), vol. 19(8), pages 1039-1051, November.
    19. Domenica Panzera & Paolo Postiglione, 2020. "Measuring the Spatial Dimension of Regional Inequality: An Approach Based on the Gini Correlation Measure," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 148(2), pages 379-394, April.
    20. Martyna Kobus, 2014. "On the measurement of polarization for ordinal data," Working Papers 325, ECINEQ, Society for the Study of Economic Inequality.

    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:zbw:iwqwdp:072015. 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: ZBW - Leibniz Information Centre for Economics (email available below). General contact details of provider: https://edirc.repec.org/data/vierlde.html .

    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.