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

Dependence Modeling

Ed. by Puccetti, Giovanni

1 Issue per year

Covered by:
SCOPUS
MathSciNet
zbMATH



Emerging Science

Open Access
Online
ISSN
2300-2298
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

Abstract

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

References

  • [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

Comments (0)

Please log in or register to comment.
Log in