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Year 2021, Volume: 42 Issue: 1, 201 - 208, 29.03.2021

Mehmet Yılmaz , Muhammet Bekçı

https://doi.org/10.17776/csj.753556

Abstract

References

  • [1] Lai C. D., Xie M., A New Family of Positive Quadrant Dependent Bivariate Distributions, Statistics and Probability Letters, 46 (4) (2000) 359-364.
  • [2] Bairamov I., Kotz S., Bekçi M., New generalized Farlie-Gumbel-Morgenstern distributions and concomitants of order statistics. Journal of Applied Statistics, 28 (5) (2001) 521-536.
  • [3] Rüschendorf L., Construction of multivariate distributions with given marginals. Annals of the Institute of Statistical Mathematics, 37 (1985) 225-233.
  • [4] Nelsen R. B., An Introduction to Copulas. Second Edition. Springer, New York (2006).
  • [5] Asadian N., Amini M., Bozorgnia A. Some concepts of negative dependence for bivariate distributions with applications. Journal of Mathematical Extension, 4 (1) (2009) 43-59.
  • [6] Hoeffding W., Masstabinvariante Korrelationstheorie, Schriften des Mathematischen Instituts und Instituts fur Angewandte Mathematik der Universitat Berlin, 5 (1940) 181-233.
  • [7] Fréchet M., Sur Les Tableaux de Corrélation Dont Les marges Sont Donnees, Annales de l’Université de Lyon, Sciences 4 (1951) 13-84.
  • [8] Schweizer B., Wolff E., On Nonparametric Measures of Dependence for Random Variables, The Annals of Statistics, 9(4) (1981) 879-885.
  • [9] Farlie D., The Performance of Some Correlation Coefficients for a General Bivariate Distribution, Biometrika, 47 (3/4) (1960) 307-323.
  • [10] Gumbel E. J., Bivariate Exponential Distributions, Journal of the American Statistical Association, 55 (292) (1960a) 698-707.
  • [11] Gumbel E. J., Bivariate Logistic Distributions, Journal of the American Statistical Association, 56 (1961) 335-349.
  • [12] Hougaard P., A Class of Multivariate Failure Time Distributions, Biometrika, 73 (1986) 671-678.
  • [13] Joe H., Multivariate Models and Dependence Concepts. Chapman and Hall, London (1997).
  • [14] Singh V.P., Zhang L. Copula–Entropy Theory for Multivariate Stochastic Modeling in Water Engineering, Geosci. Lett., 5(6) (2018).
  • [15] Frees E.W., Valdez E.A., Understanding Relationships Using Copulas, North American Actuarial Journal, 2 (1) (1998) 1-25.
  • [16] Hürlimann W., Properties and Measures of Dependence for the Archimax Copula, Advances and Applications in Statistis, 5 (2005) 125-143.
  • [17] Ali M.M., Mikhail N.N., Haq M.S., A Class of Bivariate Distributions Including the Bivariate Logistic, Journal of Multivariate Analysis, 8 (1978) 405-412.
  • [18] Balakrishnan N., Lai C. D., Continuous Bivariate Distributions. Springer Science & Business Media, (2009).
  • [19] Kumar P., Probability Distributions and Estimation of Ali-Mikhail-Haq Copula, Applied Mathematical Sciences, 4 (14) (2010) 657-666.
  • [20] Clayton D., A Model for Association in Bivariate Life Tables and its Application in Epidemiological Studies of Family Tendency in Chronic Disease Incidence, Biometrika, 65 (1978) 141-151.
  • [21] Frank M.J., On the Simultaneous Associativity of F(x,y) and x + y – F(x,y), Aequationes Math, 19 (1979) 194-226.
  • [22] Barnett V., Some Bivariate Uniform Distributions, Communications in Statistics: Theory and Methods, 9 (1980) 453-461.
  • [23] Fredricks G.A., Nelsen R.B., On the Relationship Between Spearman's Rho and Kendall's Tau for Pairs of Continuous Random Variables, Journal of Statistical Planning and Inference, 137 (7) (2007) 2143-2150.

Construction of a bivariate copula by Rüschendorf’s method

Year 2021, Volume: 42 Issue: 1, 201 - 208, 29.03.2021

Mehmet Yılmaz , Muhammet Bekçı

https://doi.org/10.17776/csj.753556

Abstract

In this paper, a new copula model with given unit marginals is proposed, based on Rüschendorf’s Method. A new bivariate copula family is introduced by adding a proper term to independence copula. Thus, we avoid the complexity of the proposed copula model. By choosing a baseline copula from the same marginal, we derive a new copula that can approach from above towards the independence copula. Furthermore, it is established that a bivariate copula constructed by this method allows some flexibility in the dependence measure according to Spearman’s correlation coefficient. Additionally, tail dependence measures are investigated. Illustrative examples are given taking into account the specific choices of a baseline copula.

Keywords

Dependence Rüschendorf’s method Bivariate copula Fréchet bounds Spearman’s rho

References

  • [1] Lai C. D., Xie M., A New Family of Positive Quadrant Dependent Bivariate Distributions, Statistics and Probability Letters, 46 (4) (2000) 359-364.
  • [2] Bairamov I., Kotz S., Bekçi M., New generalized Farlie-Gumbel-Morgenstern distributions and concomitants of order statistics. Journal of Applied Statistics, 28 (5) (2001) 521-536.
  • [3] Rüschendorf L., Construction of multivariate distributions with given marginals. Annals of the Institute of Statistical Mathematics, 37 (1985) 225-233.
  • [4] Nelsen R. B., An Introduction to Copulas. Second Edition. Springer, New York (2006).
  • [5] Asadian N., Amini M., Bozorgnia A. Some concepts of negative dependence for bivariate distributions with applications. Journal of Mathematical Extension, 4 (1) (2009) 43-59.
  • [6] Hoeffding W., Masstabinvariante Korrelationstheorie, Schriften des Mathematischen Instituts und Instituts fur Angewandte Mathematik der Universitat Berlin, 5 (1940) 181-233.
  • [7] Fréchet M., Sur Les Tableaux de Corrélation Dont Les marges Sont Donnees, Annales de l’Université de Lyon, Sciences 4 (1951) 13-84.
  • [8] Schweizer B., Wolff E., On Nonparametric Measures of Dependence for Random Variables, The Annals of Statistics, 9(4) (1981) 879-885.
  • [9] Farlie D., The Performance of Some Correlation Coefficients for a General Bivariate Distribution, Biometrika, 47 (3/4) (1960) 307-323.
  • [10] Gumbel E. J., Bivariate Exponential Distributions, Journal of the American Statistical Association, 55 (292) (1960a) 698-707.
  • [11] Gumbel E. J., Bivariate Logistic Distributions, Journal of the American Statistical Association, 56 (1961) 335-349.
  • [12] Hougaard P., A Class of Multivariate Failure Time Distributions, Biometrika, 73 (1986) 671-678.
  • [13] Joe H., Multivariate Models and Dependence Concepts. Chapman and Hall, London (1997).
  • [14] Singh V.P., Zhang L. Copula–Entropy Theory for Multivariate Stochastic Modeling in Water Engineering, Geosci. Lett., 5(6) (2018).
  • [15] Frees E.W., Valdez E.A., Understanding Relationships Using Copulas, North American Actuarial Journal, 2 (1) (1998) 1-25.
  • [16] Hürlimann W., Properties and Measures of Dependence for the Archimax Copula, Advances and Applications in Statistis, 5 (2005) 125-143.
  • [17] Ali M.M., Mikhail N.N., Haq M.S., A Class of Bivariate Distributions Including the Bivariate Logistic, Journal of Multivariate Analysis, 8 (1978) 405-412.
  • [18] Balakrishnan N., Lai C. D., Continuous Bivariate Distributions. Springer Science & Business Media, (2009).
  • [19] Kumar P., Probability Distributions and Estimation of Ali-Mikhail-Haq Copula, Applied Mathematical Sciences, 4 (14) (2010) 657-666.
  • [20] Clayton D., A Model for Association in Bivariate Life Tables and its Application in Epidemiological Studies of Family Tendency in Chronic Disease Incidence, Biometrika, 65 (1978) 141-151.
  • [21] Frank M.J., On the Simultaneous Associativity of F(x,y) and x + y – F(x,y), Aequationes Math, 19 (1979) 194-226.
  • [22] Barnett V., Some Bivariate Uniform Distributions, Communications in Statistics: Theory and Methods, 9 (1980) 453-461.
  • [23] Fredricks G.A., Nelsen R.B., On the Relationship Between Spearman's Rho and Kendall's Tau for Pairs of Continuous Random Variables, Journal of Statistical Planning and Inference, 137 (7) (2007) 2143-2150.
There are 23 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Natural Sciences
Authors

Mehmet Yılmaz 0000-0002-9762-6688

Muhammet Bekçı 0000-0001-9642-872X

Publication Date March 29, 2021
Submission Date June 16, 2020
Acceptance Date January 18, 2021
Published in Issue Year 2021Volume: 42 Issue: 1

Cite

APA Yılmaz, M., & Bekçı, M. (2021). Construction of a bivariate copula by Rüschendorf’s method. Cumhuriyet Science Journal, 42(1), 201-208. https://doi.org/10.17776/csj.753556