Abstract
In this paper, we study the dependence structure of some bivariate distribution functions based on dependence measures of Kochar and Gupta (Biometrika 74(3):664–666, 1987) and Shetty and Pandit (Stat Methods Appl 12:5–17, 2003) and then compare these measures with Spearman’s rho and Kendall’s tau. Moreover, the empirical power of the class of distribution-free tests introduced by Kochar and Gupta (1987) and Shetty and Pandit (2003) is computed based on exact and asymptotic distribution of U-statistics. Our results are obtained from simulation work in some continuous bivariate distributions for the sample of sizes \(n=6,8,15,20\) and 50. Also, we apply some examples to illustrate the results. Finally, we compare the common estimators of dependence parameter based on empirical MSE.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amini M, Jabbari H, Mohtashami Borzadaran GR, Azadbakhsh M (2010) Power comparison of independence test for the Farlie–Gumbel–Morgenstern family. Commun Korean Stat Soc 17(4):493–505
Amini M, Jabbari H, Mohtashami Borzadaran GR (2011) Aspects of dependence in generalized Farlie–Gumbel–Morgenstern distributions. Commun Stat Simul Comput 40:1192–1205
Asadian N, Amini M, Bozorgnia A (2008) Aspects of dependence in Lomax distribution. Commun Korean Stat Soc 15(2):193–204
Block HW, Basu AP (1974) A continuous bivariate exponential extension. J Am Stat Assoc 69:1031–1037
Celebioǧlu S (1997) A way of generating comprehensive copulas. J Inst Sci Technol 10(1):57–61
Cook RD, Johnson ME (1986) Generalized Burr–Pareto–Logistic distributions with applications to a uranium exploration data set. Technometrics 28(2):123–131
Cuadras CM (2009) Constructing copula functions with weighted geometric means. J Stat Plan Inference 139:3766–3772
Fasano G, Franceschini A (1987) A multidimensional version of the Kolmogorov–Smirnov test. Mon Not R Astron Soc 225:155–170
Genest C (1987) Franks family of bivariate distributions. Biometrika 74:549–555
Hutchinson TP, Lai CD (1990) Continuous bivariate distributions, emphasising applications. Rumsby Scientific Publishing, Adelaide
Kochar SG, Gupta RP (1987) Competitors of Kendall-tau test for testing independence against positive quadrant dependence. Biometrika 74(3):664–666
Kochar SG, Gupta RP (1990) Distribution-free tests based on sub-sample extrema for testing against positive dependence. Aust J Stat 32:45–51
Lehmann EL (1966) Some concepts of dependence. Ann Math Stat 37:1137–1153
Nadarajah S, Mitov K, Kotz S (2003) Local dependence functions for extreme value distributions. J Appl Stat 30(10):1081–1100
Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer, New York
Peacock JA (1983) Two-dimensional goodness-of-fit testing in astronomy. Mon Not R Astron Soc 202:615–627
Serfling RJ (1980) Approximations theorems of mathematical statistics. Wiley, New York
Shetty ID, Pandit PV (2003) Distribution-free tests for independence against positive quadrant dependence: a generalization. Stat Methods Appl 12:5–17
Sklar A (1959) Fonctions de repartition a n dimensions et leurs marges. Publications de l’institue de Statistique de l’Universitte de Paris 8:229–231
Acknowledgments
The authors would like to thank the associate editor and referees for their careful reading and constructive comments that improved presentation of the paper. The research was supported by a grant from Ferdowsi University of Mashhad (No. MS93321JAB).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zargar, M., Jabbari, H. & Amini, M. Dependence structure and test of independence for some well-known bivariate distributions. Comput Stat 32, 1423–1451 (2017). https://doi.org/10.1007/s00180-016-0696-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00180-016-0696-9