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A New Approach to Bivariate Transmutation: Construction of Continuous Bivariate Distribution Under Negative Dependency

Year 2020, Volume: 41 Issue: 4, 938 - 943, 29.12.2020

Hüseyin Ünözkan Mehmet Yılmaz

https://doi.org/10.17776/csj.742159

Abstract

In this study, a new approach to transmutation theory is developed by using negative dependency basement. Once choosing a distribution that has negative dependency with the same marginal, a new bivariate distribution is derived. In this study, we examined a new transmutation technique in which a negative dependency offers a big success in modeling rather than most known and used statistical distributions. This approach clash with classical transmutation methods. In this study at the beginning, the classical transmutation is defined. Later, we introduce the new technique and obtain lower and upper bounds of distribution to show that this approach gives us a distribution. Gaining new bivariate continuous distributions with this technique may be more appropriate in theory, and modeling of some data sets in terms of this approach may be more efficient.

Keywords

Transmuted bivariate distribution Dependence Bivariate distribution Negative dependency Fréchet bounds

References

  • [1] Dolati A., Ubeda-Flores M., Constructing Copulas by Means of Pairs of Order Statistics, Kybernetika, 45(6) (2009) 992-1002.
  • [2] Lai C. D., Xie M., A New Family of Positive Quadrant Dependent Bivariate Distributions, Statistics and Probability Letters, 46(4) (2000) 359-364.
  • [3] Kimeldorf G., Sampson A.R., A Framework for Positive Dependence, Ann. Inst. Statist. Math., 41(1) (1989) 31-45.
  • [4] Han Kwang-Hee., A New Family of Negative Quadrant Dependent Bivariate Distributions with Continuous Marginal, Journal of the Chungcheong Mathematical Society, 24 (4) (2011) 795-805.
  • [5] Shaw W.T., Buckley I.R.C., The Alchemy of Probability Distributions:Beyond Gram-Charlier & Cornish-Fisher Expansions,and Skew-Normal or Kurtotic-Normal Distributions, Financial Mathematics Group, King’s College, 1 (28) (2007).
  • [6] Aryal G.R., Tsokos C.P., On the transmuted extreme value distribution with application. Nonlinear Analysis, Theory, Methods & Applications, 71 (12) (2009) 1401-1407.
  • [7] Aryal G.R., Tsokos C.P., Transmuted Weibull Distribution, European Journal of Pure and Applied Mathematics, 4 (2) (2011) 89-102.
  • [8] Merovci F., Transmuted Lindley Distribution, Int. J. Open Problems Compt. Math., 6 (2) (2013) 63-72.
  • [9] Merovci F., Transmuted Exponentiated Exponential Distribution, Mathematical Sciences and Applications E-Notes, 1 (2) (2013) 112-122.
  • [10] Merovci F., Transmuted Rayleigh Distribution, Austrian Journal of Statistics, 42 (1) (2013) 21-31.
  • [11] Aryal G.R., Transmuted Log-Logistic Distribution, Journal of Statistics Applications & Probability, 2 (1) (2013) 11-20.
  • [12] Elbatal I., Diab L.S. and AbdulAlim N.A., Transmuted Generalized Linear Exponential Distribution, International Journal of Computer Applications, 83 (17) 82013) 29-37.
  • [13] Merovci F., Puka L., Transmuted Pareto distribution, ProbStat Forum, 7 (2014) 1-11.
  • [14] Mohamed H.A., Transmuted Exponentiated Gamma Distribution: A Generalization of the Exponentiated Gamma Probability Distribution, Applied Mathematical Sciences, 8 (27) (2014) 1297-1310.
  • [15] Oguntunde E.P., Adejumo O.A., The Transmuted Inverse Exponential Distribution, International Journal of Advanced Statistics and Probability, 3 (1) 82015) 1-7.
  • [16] Ünözkan H., Yılmaz M., A New Method for Generating Distributions: An Application to Flow Data, International Journal of Statistics and Applications, 9 (3) (2019) 92-99
  • [17] Barlow R. E., Proschan, F., Statistical Theory of Reliability and Life Testing: Probability Models, Technical Report, New York: Holt, Rinehart and Winston, 1975.
  • [18] Domma F., Bivariate Reversed Hazard Rate, Notions, and Measures of Dependence and their Relationships, Communications in Statistics - Theory and Methods, 40(6) (2011) 989-999, DOI: 10.1080/03610920903511777.
  • [19] Farlie D., The Performance of Some Correlation Coefficients for a General Bivariate Distribution, Biometrika, 47(3/4) (1960) 307-323.
  • [20] Gumbel E. J., Bivariate Exponential Distributions, Journal of the American Statistical Association, 55-292 (1960) 698-707.
  • [21] Bismi G., Bivariate Burr Distributions, unpublished PhD Thesis, Cochin University of Science and Technology, 2005.
  • [22] Basu A.P., Bivariate Failure Rate, Journal of the American Statistical Association, 66 (1971) 103–104.
  • [23] Hoeffding W., Masstabinvariante Korrelationstheorie, Schriften des Mathematischen Instituts und Instituts fur Angewandte Mathematik der Universitat Berlin, 5 (1940) 181-233.
  • [24] Fréchet M., Sur Les Tableaux de Corrélation Dont Les marges Sont Donnees, Annales de l’Université de Lyon, Sciences, 4 (1951) 13–84.
  • [25] Schweizer B. and Wolff E., On Nonparametric Measures of Dependence for Random Variables, The Annals of Statistics, 9(4) (1981) 879-885.

Year 2020, Volume: 41 Issue: 4, 938 - 943, 29.12.2020

Hüseyin Ünözkan Mehmet Yılmaz

https://doi.org/10.17776/csj.742159

Abstract

References

  • [1] Dolati A., Ubeda-Flores M., Constructing Copulas by Means of Pairs of Order Statistics, Kybernetika, 45(6) (2009) 992-1002.
  • [2] Lai C. D., Xie M., A New Family of Positive Quadrant Dependent Bivariate Distributions, Statistics and Probability Letters, 46(4) (2000) 359-364.
  • [3] Kimeldorf G., Sampson A.R., A Framework for Positive Dependence, Ann. Inst. Statist. Math., 41(1) (1989) 31-45.
  • [4] Han Kwang-Hee., A New Family of Negative Quadrant Dependent Bivariate Distributions with Continuous Marginal, Journal of the Chungcheong Mathematical Society, 24 (4) (2011) 795-805.
  • [5] Shaw W.T., Buckley I.R.C., The Alchemy of Probability Distributions:Beyond Gram-Charlier & Cornish-Fisher Expansions,and Skew-Normal or Kurtotic-Normal Distributions, Financial Mathematics Group, King’s College, 1 (28) (2007).
  • [6] Aryal G.R., Tsokos C.P., On the transmuted extreme value distribution with application. Nonlinear Analysis, Theory, Methods & Applications, 71 (12) (2009) 1401-1407.
  • [7] Aryal G.R., Tsokos C.P., Transmuted Weibull Distribution, European Journal of Pure and Applied Mathematics, 4 (2) (2011) 89-102.
  • [8] Merovci F., Transmuted Lindley Distribution, Int. J. Open Problems Compt. Math., 6 (2) (2013) 63-72.
  • [9] Merovci F., Transmuted Exponentiated Exponential Distribution, Mathematical Sciences and Applications E-Notes, 1 (2) (2013) 112-122.
  • [10] Merovci F., Transmuted Rayleigh Distribution, Austrian Journal of Statistics, 42 (1) (2013) 21-31.
  • [11] Aryal G.R., Transmuted Log-Logistic Distribution, Journal of Statistics Applications & Probability, 2 (1) (2013) 11-20.
  • [12] Elbatal I., Diab L.S. and AbdulAlim N.A., Transmuted Generalized Linear Exponential Distribution, International Journal of Computer Applications, 83 (17) 82013) 29-37.
  • [13] Merovci F., Puka L., Transmuted Pareto distribution, ProbStat Forum, 7 (2014) 1-11.
  • [14] Mohamed H.A., Transmuted Exponentiated Gamma Distribution: A Generalization of the Exponentiated Gamma Probability Distribution, Applied Mathematical Sciences, 8 (27) (2014) 1297-1310.
  • [15] Oguntunde E.P., Adejumo O.A., The Transmuted Inverse Exponential Distribution, International Journal of Advanced Statistics and Probability, 3 (1) 82015) 1-7.
  • [16] Ünözkan H., Yılmaz M., A New Method for Generating Distributions: An Application to Flow Data, International Journal of Statistics and Applications, 9 (3) (2019) 92-99
  • [17] Barlow R. E., Proschan, F., Statistical Theory of Reliability and Life Testing: Probability Models, Technical Report, New York: Holt, Rinehart and Winston, 1975.
  • [18] Domma F., Bivariate Reversed Hazard Rate, Notions, and Measures of Dependence and their Relationships, Communications in Statistics - Theory and Methods, 40(6) (2011) 989-999, DOI: 10.1080/03610920903511777.
  • [19] Farlie D., The Performance of Some Correlation Coefficients for a General Bivariate Distribution, Biometrika, 47(3/4) (1960) 307-323.
  • [20] Gumbel E. J., Bivariate Exponential Distributions, Journal of the American Statistical Association, 55-292 (1960) 698-707.
  • [21] Bismi G., Bivariate Burr Distributions, unpublished PhD Thesis, Cochin University of Science and Technology, 2005.
  • [22] Basu A.P., Bivariate Failure Rate, Journal of the American Statistical Association, 66 (1971) 103–104.
  • [23] Hoeffding W., Masstabinvariante Korrelationstheorie, Schriften des Mathematischen Instituts und Instituts fur Angewandte Mathematik der Universitat Berlin, 5 (1940) 181-233.
  • [24] Fréchet M., Sur Les Tableaux de Corrélation Dont Les marges Sont Donnees, Annales de l’Université de Lyon, Sciences, 4 (1951) 13–84.
  • [25] Schweizer B. and Wolff E., On Nonparametric Measures of Dependence for Random Variables, The Annals of Statistics, 9(4) (1981) 879-885.
There are 25 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Natural Sciences
Authors

Hüseyin Ünözkan 0000-0001-9659-287X

Mehmet Yılmaz 0000-0002-9762-6688

Publication Date December 29, 2020
Submission Date May 24, 2020
Acceptance Date August 19, 2020
Published in Issue Year 2020Volume: 41 Issue: 4

Cite

APA Ünözkan, H., & Yılmaz, M. (2020). A New Approach to Bivariate Transmutation: Construction of Continuous Bivariate Distribution Under Negative Dependency. Cumhuriyet Science Journal, 41(4), 938-943. https://doi.org/10.17776/csj.742159