Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Dependence Modeling

Ed. by Puccetti, Giovanni

1 Issue per year

Covered by:

Emerging Science

Open Access
See all formats and pricing
More options …

Dependence Measuring from Conditional Variances

Noppadon Kamnitui
  • Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Tippawan Santiwipanont
  • Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Songkiat Sumetkijakan
  • Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-07-22 | DOI: https://doi.org/10.1515/demo-2015-0007


A conditional variance is an indicator of the level of independence between two random variables. We exploit this intuitive relationship and define a measure v which is almost a measure of mutual complete dependence. Unsurprisingly, the measure attains its minimum value for many pairs of non-independent ran- dom variables. Adjusting the measure so as to make it invariant under all Borel measurable injective trans- formations, we obtain a copula-based measure of dependence v* satisfying A. Rényi’s postulates. Finally, we observe that every nontrivial convex combination of v and v* is a measure of mutual complete dependence.

Keywords: conditional variances; measures of dependence; copulas; mutual complete dependence; shuffles of Min

MSC:: 60A10; 62H20


  • [1] V.I. Bogachev, Measure Theory, vol I, Springer Verlag, 2007. Google Scholar

  • [2] N. Chaidee, T. Santiwipanont, S. Sumetkijakan, Patched approximations and their convergence, Comm. Statist. Theory Methods, in press, http://dx.doi.org/10.1080/03610926.2014.887112. CrossrefGoogle Scholar

  • [3] W.F. Darsow, B. Nguyen, E.T. Olsen, Copulas and Markov processes, Illinois J. Math. 36 (1992) 600–642. Google Scholar

  • [4] W.F. Darsow, E.T. Olsen, Norms for copulas, Int. J. Math. Math. Sci. 18 (1995) 417–436. CrossrefGoogle Scholar

  • [5] W.F. Darsow, E.T. Olsen, Characterization of idempotent 2-copulas, Note Mat. 30 (2010) 147–177. Google Scholar

  • [6] F. Durante, E.P. Klement, J.J. Quesada-Molina, P. Sarkoci, Remarks on two product-like constructions for copulas, Kyber- netika (Prague) 43 (2007) 235–244. Google Scholar

  • [7] F. Durante, P. Sarkoci, C. Sempi, Shuffles of copulas, J. Math. Anal. Appl. 352 (2009) 914–921. Google Scholar

  • [8] H. Gebelein, Das statistische Problem der Korrelation als Variations und Eigenwertproblem und sein Zusammenhang mit der Ausgleichsrechnung, Z. Angew. Math. Mech. 21 (1941) 364–379. CrossrefGoogle Scholar

  • [9] N. Kamnitui, New Measure of Dependence from Conditional Variances, Master thesis, 2015. Google Scholar

  • [10] H.O. Lancaster, Correlation and complete dependence of random variables, Ann. Math. Statist. 34 (1963) 1315–1321. CrossrefGoogle Scholar

  • [11] P. Mikusiński, H. Sherwood, M.D. Taylor, Probabilistic interpretations of copulas and their convex sums, in: G. Dall’Aglio, S. Kotz, G. Salinetti (Eds.), Advances in Probability Distributionswith GivenMarginals: Beyond the Copulas, Kluwer Dordrecht. 67 (1991) 95–112. Google Scholar

  • [12] P. Mikusiński, H. Sherwood, M.D. Taylor, Shuffles of min, Stochastica 13 (1992) 61–74. Google Scholar

  • [13] R.B. Nelsen, An Introduction to Copulas, second ed., Springer Verlag, 2006. Google Scholar

  • [14] K. Pearson, D. Heron, On theories of association, Biometrika 9(1/2) (1913) 159–315. CrossrefGoogle Scholar

  • [15] E.T. Olsen, W.F. Darsow, B. Nguyen, Copulas and Markov operators, Lecture Notes-Monograph Series 28 (1996) 244–259. Google Scholar

  • [16] A. Rényi, On measures of dependence, Acta. Math. Acad. Sci. Hungar. 10 (1959) 441–451. CrossrefGoogle Scholar

  • [17] P. Ruankong, T. Santiwipanont, S. Sumetkijakan, Shuffles of copulas and a new measure of dependence, J.Math. Anal. Appl. 398(1) (2013) 398–402. Google Scholar

  • [18] B. Schweizer, E.F. Wolff, On nonparametric measures of dependence for random variables, Ann. Statist. 9 (1981) 879–885. CrossrefGoogle Scholar

  • [19] K.F. Siburg, P.A. Stoimenov, A scalar product for copulas, J. Math. Anal. Appl. 344 (2008) 429–439. Google Scholar

  • [20] K.F. Siburg, P.A. Stoimenov, A measure of mutual complete dependence, Metrika 71 (2009) 239–251. CrossrefWeb of ScienceGoogle Scholar

  • [21] A. Sklar, Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8 (1959) 229–231. Google Scholar

  • [22] W. Trutschnig, On a strong metric on the space of copulas and its induced dependence measure, J. Math. Anal. Appl. 384 (2011) 690–705. Web of ScienceGoogle Scholar

  • [23] W. Trutschnig, On Cesaro convergence of iterates of the star product of copulas, Stat. Prob. Letters 83 (2013) 357–365. Web of ScienceCrossrefGoogle Scholar

  • [24] Y. Zheng, J. Yang, J.Z. Huang, Approximation of bivariate copulas by patched bivariate Fréchet copulas, Insurance Math. Econ. 48 (2011) 246–256. CrossrefGoogle Scholar

About the article

Received: 2015-02-28

Accepted: 2015-07-02

Published Online: 2015-07-22

Citation Information: Dependence Modeling, Volume 3, Issue 1, ISSN (Online) 2300-2298, DOI: https://doi.org/10.1515/demo-2015-0007.

Export Citation

© 2015 N. Kamnitui et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Nadezhda Gribkova and Ričardas Zitikis
Annals of the Institute of Statistical Mathematics, 2018

Comments (0)

Please log in or register to comment.
Log in