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

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  • 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
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    1. 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.
    2. Micheas, Athanasios C. & Zografos, Konstantinos, 2006. "Measuring stochastic dependence using [phi]-divergence," Journal of Multivariate Analysis, Elsevier, vol. 97(3), pages 765-784, March.
    3. de Paula, Aureo, 2008. "Conditional Moments and Independence," The American Statistician, American Statistical Association, vol. 62, pages 219-221, August.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. Rafael S. Calsaverini & Renato Vicente, 2009. "An information theoretic approach to statistical dependence: copula information," Papers 0911.4207, arXiv.org.
    15. 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.
    16. 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.
    17. 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.
    18. 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.
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    2. 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.
    3. 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.

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