International Journal of Nonlinear Analysis and Applications

Properties of measures of association within an extended FGM

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Education, University of Kufa, Iraq

2 General Directorate of Education in Najaf, Iraq

Abstract

In this paper, we propose an extension to the bivariate FGM copula within a polynomial function of degree one. The desired extension depends on the modification that was shown by Sriboonchitta-Kreinovich [12]. We also illustrate a general form of such extension with degree n. We examine various necessary and sufficient conditions which prove that the illustrated function within the extension is the copula. Eventually, we present several calculations of the most popular dependencies within the proposed FGM copula of degree one.

Keywords

[1] N. Blomqvist, On a measure of dependence between two random variables, Ann. Math. Statist. 21 (1950) 593–600.
[2] G.M. D’ESTE, A Morgenstern-type bivariate gamma distribution, Biometrika. 68(1) (1981) 339–340
[3] P. Deheuvels, A Kolmogorov-Smirnov type test for independence and multivariate samples, Rev. Roumaine Math.
Pures Appl. 26 (1981a) 213–226.
[4] D.J.G. Farlie, The performance of some correlation coefficients for a general bivariate distribution, Biometrika
47 (1960) 307–323.
[? ] L. Herbach, Introduction, Gumbel Model, In: de Oliveira J.T. (eds) Statistical Extremes and Applications.
NATO ASI Series (Series C: Mathematical and Physical Sciences), vol 131. Springer, Dordrecht. 1984.
[5] W. Hoeffding, A Non-Parametric Test of Independence, Ann. Math. Statist. 19(4) (1948) 546–557.
[6] M. Hollander and D.A. Wolfe, Nonparametric Statistical Methods, Wiley, New York, 1973.
[7] W.H. Kruskal, Ordinal measures of association, J. Amer. Statist. Assoc. 53 (1958) 814–861.
[8] R.B. Nelsen, J.J. Quesada Molina and J.A. Rodr´ıguez Lallena, Bivariate copulas with cubic sections, J. Nonparametr Statist. 7 (1997) 205–220.
[9] R.B. Nelsen, An Introduction to Copulas, Springer Verlag, Berlin, Heidelberg, New York, 2007.
[10] J.J. Quesada-Molina and J.A. Rodr´ıguez-Lallena, Bivariate copulas with quadratic sections, J. Nonparametric
Statist. 5(4) (1995) 323–337.
[11] J.A. Rodr´ıguez-Lallena and M. Ubeda-Flores, ´ Multivariate copulas with quadratic sections in one variable, Metrika
72(3) (2010).
[12] A. Sklar, Random variables, joint distribution functions, and copulas, Kybernetika 9 (1973) 449–460.
[13] S. Sriboonchitta and V. Kreinovich, Why are FGM copulas successful: a simple explanation, Comput. Sci. Commons 3 (2017).
[14] B. Schweizer, E.F. Wolff, On nonparametric measures of dependence for random variables, Ann. Statist. 9 (1981)
879–885.
[15] Sh. Sharma and A.J. Obaid, Mathematical modelling, analysis and design of fuzzy logic controller for the control
of ventilation systems using MATLAB fuzzy logic toolbox, J. Interdiscip. Math. 23(4) (2020) 843–849.
International Journal of Nonlinear Analysis and Applications
Volume 12, Special Issue
December 2021
Pages 2265-2272
  • Receive Date: 04 October 2021
  • Revise Date: 11 November 2021
  • Accept Date: 03 December 2021