IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v131y2014icp32-50.html
   My bibliography  Save this article

Comparison, utility, and partition of dependence under absolutely continuous and singular distributions

Author

Listed:
  • Ebrahimi, Nader
  • Jalali, Nima Y.
  • Soofi, Ehsan S.

Abstract

This paper first illustrates that a mutual information index detects and ranks dependence of a wide variety of absolutely continuous families, but the popular association and variance reduction indices fail to serve as such “common metrics”. We then elaborate on some theoretical merits of the mutual information and give several results. The mutual information provides a notion of the utility of dependence for predicting random variables and quantifies how much the joint distribution is more informative about the variables than the independent model. We present insightful partitions of dependence among the components of a random vector, for a class of models recently proposed for dependence of uncorrelated variables, and for the elliptical families. We also recall that the mutual information is not applicable to singular distributions and give some results for a generalized information index for these models. The generalized index is derived for the Marshall–Olkin copula and for a new singular copula that represents the dependence of the consecutive terms of the exponential autoregressive and related processes.

Suggested Citation

  • Ebrahimi, Nader & Jalali, Nima Y. & Soofi, Ehsan S., 2014. "Comparison, utility, and partition of dependence under absolutely continuous and singular distributions," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 32-50.
    • Handle: RePEc:eee:jmvana:v:131:y:2014:i:c:p:32-50
    DOI: 10.1016/j.jmva.2014.06.014
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X14001444
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2014.06.014?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. Nader Ebrahimi & Ehsan S. Soofi & Refik Soyer, 2010. "Information Measures in Perspective," International Statistical Review, International Statistical Institute, vol. 78(3), pages 383-412, December.
    2. Rafael S. Calsaverini & Renato Vicente, 2009. "An information theoretic approach to statistical dependence: copula information," Papers 0911.4207, arXiv.org.
    3. Hirschberg, Joseph G. & Maasoumi, Esfandiar & Slottje, Daniel J., 1991. "Cluster analysis for measuring welfare and quality of life across countries," Journal of Econometrics, Elsevier, vol. 50(1-2), pages 131-150, October.
    4. deB. Edwardes, Michael D., 1993. "Kendall's [tau] is equal to the correlation coefficient for the BVE distribution," Statistics & Probability Letters, Elsevier, vol. 17(5), pages 415-419, August.
    5. Bertrand Clarke & Jennifer Clarke & Chi Wai Yu, 2014. "Statistical Problem Classes and Their Links to Information Theory," Econometric Reviews, Taylor & Francis Journals, vol. 33(1-4), pages 337-371, June.
    6. Nader Ebrahimi & Nima Y. Jalali & Ehsan S. Soofi & Refik Soyer, 2014. "Importance of Components for a System," Econometric Reviews, Taylor & Francis Journals, vol. 33(1-4), pages 395-420, June.
    7. Robert T. Clemen & Gregory W. Fischer & Robert L. Winkler, 2000. "Assessing Dependence: Some Experimental Results," Management Science, INFORMS, vol. 46(8), pages 1100-1115, August.
    8. Micheas, Athanasios C. & Zografos, Konstantinos, 2006. "Measuring stochastic dependence using [phi]-divergence," Journal of Multivariate Analysis, Elsevier, vol. 97(3), pages 765-784, March.
    9. de Paula, Aureo, 2008. "Conditional Moments and Independence," The American Statistician, American Statistical Association, vol. 62, pages 219-221, August.
    10. Ebrahimi, Nader & Hamedani, G.G. & Soofi, Ehsan S. & Volkmer, Hans, 2010. "A class of models for uncorrelated random variables," Journal of Multivariate Analysis, Elsevier, vol. 101(8), pages 1859-1871, September.
    11. C. W. Granger & E. Maasoumi & J. Racine, 2004. "A Dependence Metric for Possibly Nonlinear Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(5), pages 649-669, September.
    12. Ebrahimi, Nader & Soofi, Ehsan S. & Soyer, Refik, 2008. "Multivariate maximum entropy identification, transformation, and dependence," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1217-1231, July.
    13. José M. Bernardo & Raúl Rueda, 2002. "Bayesian Hypothesis Testing: a Reference Approach," International Statistical Review, International Statistical Institute, vol. 70(3), pages 351-372, December.
    14. Zografos, K. & Nadarajah, S., 2005. "Expressions for Rényi and Shannon entropies for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 71(1), pages 71-84, January.
    15. Fang, Hong-Bin & Fang, Kai-Tai & Kotz, Samuel, 2002. "The Meta-elliptical Distributions with Given Marginals," Journal of Multivariate Analysis, Elsevier, vol. 82(1), pages 1-16, July.
    16. Smith, Michael & Min, Aleksey & Almeida, Carlos & Czado, Claudia, 2010. "Modeling Longitudinal Data Using a Pair-Copula Decomposition of Serial Dependence," Journal of the American Statistical Association, American Statistical Association, vol. 105(492), pages 1467-1479.
    17. Peter J. Danaher & Michael S. Smith, 2011. "Modeling Multivariate Distributions Using Copulas: Applications in Marketing," Marketing Science, INFORMS, vol. 30(1), pages 4-21, 01-02.
    18. Ebrahimi, Nader & Kirmani, S.N.U.A. & Soofi, Ehsan S., 2007. "Multivariate dynamic information," Journal of Multivariate Analysis, Elsevier, vol. 98(2), pages 328-349, February.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Asadi, Majid & Ebrahimi, Nader & Soofi, Ehsan S. & Zohrevand, Younes, 2016. "Jensen–Shannon information of the coherent system lifetime," Reliability Engineering and System Safety, Elsevier, vol. 156(C), pages 244-255.
    2. Borgonovo, Emanuele & Hazen, Gordon B. & Jose, Victor Richmond R. & Plischke, Elmar, 2021. "Probabilistic sensitivity measures as information value," European Journal of Operational Research, Elsevier, vol. 289(2), pages 595-610.
    3. Plischke, Elmar & Borgonovo, Emanuele, 2019. "Copula theory and probabilistic sensitivity analysis: Is there a connection?," European Journal of Operational Research, Elsevier, vol. 277(3), pages 1046-1059.

    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. Majid Asadi & Karthik Devarajan & Nader Ebrahimi & Ehsan Soofi & Lauren Spirko‐Burns, 2022. "Elaboration Models with Symmetric Information Divergence," International Statistical Review, International Statistical Institute, vol. 90(3), pages 499-524, December.
    2. Smith, Michael Stanley, 2023. "Implicit Copulas: An Overview," Econometrics and Statistics, Elsevier, vol. 28(C), pages 81-104.
    3. Michael Stanley Smith, 2021. "Implicit Copulas: An Overview," Papers 2109.04718, arXiv.org.
    4. Navarro, J. & Sunoj, S.M. & Linu, M.N., 2011. "Characterizations of bivariate models using dynamic Kullback-Leibler discrimination measures," Statistics & Probability Letters, Elsevier, vol. 81(11), pages 1594-1598, November.
    5. Stöber, Jakob & Hong, Hyokyoung Grace & Czado, Claudia & Ghosh, Pulak, 2015. "Comorbidity of chronic diseases in the elderly: Patterns identified by a copula design for mixed responses," Computational Statistics & Data Analysis, Elsevier, vol. 88(C), pages 28-39.
    6. Nader Ebrahimi & S.N.U.A. Kirmani & Ehsan S. Soofi, 2011. "Predictability of operational processes over finite horizon," Naval Research Logistics (NRL), John Wiley & Sons, vol. 58(6), pages 531-545, September.
    7. Maryam Eskandarzadeh & Antonio Di Crescenzo & Saeid Tahmasebi, 2019. "Cumulative Measure of Inaccuracy and Mutual Information in k -th Lower Record Values," Mathematics, MDPI, vol. 7(2), pages 1-19, February.
    8. Dißmann, J. & Brechmann, E.C. & Czado, C. & Kurowicka, D., 2013. "Selecting and estimating regular vine copulae and application to financial returns," Computational Statistics & Data Analysis, Elsevier, vol. 59(C), pages 52-69.
    9. José Antonio Rodríguez Martín & Juan Dios Jiménez Aguilera & José Antonio Salinas Fernández & José María Martín Martín, 2016. "Millennium Development Goals 4 and 5: Progress in the Least Developed Countries of Asia," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 129(2), pages 489-504, November.
    10. Janani Sri S. & Parthajit Kayal & G. Balasubramanian, 2022. "Can Equity be Safe-haven for Investment?," Journal of Emerging Market Finance, Institute for Financial Management and Research, vol. 21(1), pages 32-63, March.
    11. Vexler, Albert & Zou, Li, 2022. "Linear projections of joint symmetry and independence applied to exact testing treatment effects based on multidimensional outcomes," Journal of Multivariate Analysis, Elsevier, vol. 190(C).
    12. Jesus Perez-Mayo, 2005. "Identifying deprivation profiles in Spain: a new approach," Applied Economics, Taylor & Francis Journals, vol. 37(8), pages 943-955.
    13. Antonio Di Crescenzo & Patrizia Di Gironimo, 2018. "Stochastic Comparisons and Dynamic Information of Random Lifetimes in a Replacement Model," Mathematics, MDPI, vol. 6(10), pages 1-13, October.
    14. Carol Alexander & José María Sarabia, 2012. "Quantile Uncertainty and Value‐at‐Risk Model Risk," Risk Analysis, John Wiley & Sons, vol. 32(8), pages 1293-1308, August.
    15. Lu Yang & Claudia Czado, 2022. "Two‐part D‐vine copula models for longitudinal insurance claim data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(4), pages 1534-1561, December.
    16. Dionisio, Andreia & Menezes, Rui & Mendes, Diana & Vidigal Da Silva, Jacinto, 2007. "Nonlinear Dynamics Within Macroeconomic Factors And Stock Market In Portugal, 1993-2003," Applied Econometrics and International Development, Euro-American Association of Economic Development, vol. 7(2), pages 57-70.
    17. Agbeyegbe, Terence D., 2015. "An inverted U-shaped crude oil price return-implied volatility relationship," Review of Financial Economics, Elsevier, vol. 27(C), pages 28-45.
    18. Shilan Li & Jianxin Shi & Paul Albert & Hong-Bin Fang, 2022. "Dependence Structure Analysis and Its Application in Human Microbiome," Mathematics, MDPI, vol. 11(1), pages 1-14, December.
    19. Li, J. & Nott, D.J. & Fan, Y. & Sisson, S.A., 2017. "Extending approximate Bayesian computation methods to high dimensions via a Gaussian copula model," Computational Statistics & Data Analysis, Elsevier, vol. 106(C), pages 77-89.
    20. Luca Bagnato & Valerio Potì & Maria Zoia, 2015. "The role of orthogonal polynomials in adjusting hyperpolic secant and logistic distributions to analyse financial asset returns," Statistical Papers, Springer, vol. 56(4), pages 1205-1234, November.

    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:jmvana:v:131:y:2014:i:c:p:32-50. 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.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    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.