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Tytuł artykułu

Dependence Measuring from Conditional Variances

Treść / Zawartość
Warianty tytułu
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.
Wydawca
Czasopismo
Rocznik
Tom
3
Numer
1
Opis fizyczny
Daty
otrzymano
2015-02-28
zaakceptowano
2015-07-02
online
2015-07-22
Twórcy
  • Department of Mathematics and Computer Science, Faculty of Science,
    Chulalongkorn University, Bangkok 10330, Thailand
  • Department of Mathematics and Computer Science, Faculty of Science,
    Chulalongkorn University, Bangkok 10330, Thailand
  • 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. TheoryMethods, 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 mitder 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]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_demo-2015-0007
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