Li, Hui (2016): A true measure of dependence.

PDF
MPRA_paper_69735.pdf Download (129kB)  Preview 
Abstract
The strength of dependence between random variables is an important property that is useful in a lot of areas. Various measures have been proposed which detect mostly divergence from independence. However, a true measure of dependence should also be able to characterize complete dependence where one variable is a function of the other. Previous measures are mostly symmetric which are shown to be insufficient to capture complete dependence. A new type of nonsymmetric dependence measure is presented that can unambiguously identify both independence and complete dependence. The original Rényi’s axioms for symmetric measures are reviewed and modified for nonsymmetric measures.
Item Type:  MPRA Paper 

Original Title:  A true measure of dependence 
Language:  English 
Keywords:  Nonsymmetric dependence measure, complete dependence, ∗ product on copula, Data Processing Inequality (DPI) 
Subjects:  C  Mathematical and Quantitative Methods > C0  General > C02  Mathematical Methods 
Item ID:  69735 
Depositing User:  Hui Li 
Date Deposited:  11 Apr 2016 05:43 
Last Modified:  26 Sep 2019 18:13 
References:  1. Cover, T. M. and Thomas, J. A. (1991). Elements of information theory. New York: John Wiley & Sons. 2. Darsow, W. F., Nguyen, B. and Olsen, E. T. (1992). Copulas and Markov processes. Illinois J. Math. 36 600642. 3. Dette, H., Siburg, K. F. and Stoimenov, P. A. (2010). A copulabased nonparametric measure of regression dependence. Scand. J. Stat. 40 2141. 4. Granger, C. W., Maasoumi, E. and Racine, J. (2004). A dependence metric for possibly nonlinear processes. J. Time Ser. Anal. 25 649669. 5. HlaváčkováSchindler, K., Paluš, M., Vejmelka, M. and Bhattacharya, J. (2007). Causality detection based on informationtheoretic approaches in time series analysis. Phys. Rep. 441 146. 6. Jaworski, P., Durante, F., Härdle, W. and Rychlik, T. (Eds.) (2010). Copula Theory and Its Applications. Heidelberg: Springer. 7. Kinney, J. B. and Atwal, G. S. (2014). Equitability, mutual information and the maximal information coefficient. Proc. Nat. Acad. Sci. 111 33543359. 8. Li, H. (2015a). On Nonsymmetric Nonparametric Measures of Dependence. arXiv:1502.03850. 9. Li, H. (2015b). A new class of nonsymmetric multivariate dependence measures. arXiv:1511.02744. 10. Li, H. (2015c). Nonsymmetric dependence measures: the discrete case. arXiv:1512.07945. 11. Linfoot, E. H. (1957). An informational measure of correlation. Information and Control 1 8589. 12. Nelson, R. B. (2006). An introduction to copulas. New York: Springer. 13. Rényi, A. (1959). On measures of dependence. Acta Math. Acad. Sci. Hungar. 10 441451. 14. Rényi, A. (1961). On measures of entropy and information. In: Proceedings of the Fourth Berkley Symposium on Mathematical Statistics and Probability 1 547561. 15. Shannon, C. E. and Weaver, W. (1949). The mathematical theory of communication. Urbana: University of Illinois Press. 16. Sklar, M. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 229231. 17. Tsallis, C. (1988). Possible generalization of BoltzmannGibbs statistics. J. Stat. Phys. 52 479487. 18. Trutschnig, W. (2011). On a strong metric on the space of copulas and its induced dependence measure. J. Math. Anal. Appl. 384 690705. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/69735 