Dependence Measuring from Conditional Variances

• Autorzy: Noppadon Kamnitui , Tippawan Santiwipanont , Songkiat Sumetkijakan
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

conditional variances; measures of dependence; copulas; mutual complete dependence; shufflesof Min

Kategorie tematyczne

• 62H20: Measures of association (correlation, canonical correlation, etc.)
• 60A10: Probabilistic measure theory

• Wydawca: De Gruyter Open
• Czasopismo: Dependence Modeling
• Rocznik: 2015
• Tom: 3
• Numer: 1

Daty

otrzymano

2015-02-28

zaakceptowano

2015-07-02

online

2015-07-22

Twórcy

autor

Noppadon Kamnitui

• Department of Mathematics and Computer Science, Faculty of Science,
  Chulalongkorn University, Bangkok 10330, Thailand

autor

Tippawan Santiwipanont

• Department of Mathematics and Computer Science, Faculty of Science,
  Chulalongkorn University, Bangkok 10330, Thailand

autor

Songkiat Sumetkijakan

• Department of Mathematics and Computer Science, Faculty of Science,
  Chulalongkorn University, Bangkok 10330, Thailand

Bibliografia

• [1] V.I. Bogachev, Measure Theory, vol I, Springer Verlag, 2007.
• [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. [Crossref]
• [3] W.F. Darsow, B. Nguyen, E.T. Olsen, Copulas and Markov processes, Illinois J. Math. 36 (1992) 600–642.
• [4] W.F. Darsow, E.T. Olsen, Norms for copulas, Int. J. Math. Math. Sci. 18 (1995) 417–436. [Crossref]
• [5] W.F. Darsow, E.T. Olsen, Characterization of idempotent 2-copulas, Note Mat. 30 (2010) 147–177.
• [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.
• [7] F. Durante, P. Sarkoci, C. Sempi, Shuffles of copulas, J. Math. Anal. Appl. 352 (2009) 914–921.
• [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. [Crossref]
• [9] N. Kamnitui, New Measure of Dependence from Conditional Variances, Master thesis, 2015.
• [10] H.O. Lancaster, Correlation and complete dependence of random variables, Ann. Math. Statist. 34 (1963) 1315–1321. [Crossref]
• [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.
• [12] P. Mikusiński, H. Sherwood, M.D. Taylor, Shuffles of min, Stochastica 13 (1992) 61–74.
• [13] R.B. Nelsen, An Introduction to Copulas, second ed., Springer Verlag, 2006.
• [14] K. Pearson, D. Heron, On theories of association, Biometrika 9(1/2) (1913) 159–315. [Crossref]
• [15] E.T. Olsen, W.F. Darsow, B. Nguyen, Copulas and Markov operators, Lecture Notes-Monograph Series 28 (1996) 244–259.
• [16] A. Rényi, On measures of dependence, Acta. Math. Acad. Sci. Hungar. 10 (1959) 441–451. [Crossref]
• [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.
• [18] B. Schweizer, E.F. Wolff, On nonparametric measures of dependence for random variables, Ann. Statist. 9 (1981) 879–885. [Crossref]
• [19] K.F. Siburg, P.A. Stoimenov, A scalar product for copulas, J. Math. Anal. Appl. 344 (2008) 429–439.
• [20] K.F. Siburg, P.A. Stoimenov, A measure of mutual complete dependence, Metrika 71 (2009) 239–251. [Crossref][WoS]
• [21] A. Sklar, Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8 (1959) 229–231.
• [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. [WoS]
• [23] W. Trutschnig, On Cesaro convergence of iterates of the star product of copulas, Stat. Prob. Letters 83 (2013) 357–365. [WoS][Crossref]
• [24] Y. Zheng, J. Yang, J.Z. Huang, Approximation of bivariate copulas by patched bivariate Fréchet copulas, Insurance Math. Econ. 48 (2011) 246–256. [Crossref]

Identyfikatory

DOI

10.1515/demo-2015-0007