Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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Comparing the empirical powers of several independence tests in generalized FGM family

  • Zargar, M. (Department of Statistics, Ferdowsi University of Mashhad) ;
  • Jabbari, H. (Department of Statistics, Ferdowsi University of Mashhad) ;
  • Amini, M. (Department of Statistics, Ferdowsi University of Mashhad)
  • Received : 2016.03.01
  • Accepted : 2016.05.03
  • Published : 2016.05.31
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Abstract

The powers of some tests for independence hypothesis against positive (negative) quadrant dependence in generalized Farlie-Gumbel-Morgenstern distribution are compared graphically by simulation. Some of these tests are usual linear rank tests of independence. Two other possible rank tests of independence are locally most powerful rank test and a powerful nonparametric test based on the $Cram{\acute{e}}r-von$ Mises statistic. We also evaluate the empirical power of the class of distribution-free tests proposed by Kochar and Gupta (1987) based on the asymptotic distribution of a U-statistic and the test statistic proposed by $G{\ddot{u}}ven$ and Kotz (2008) in generalized Farlie-Gumbel-Morgenstern distribution. Tests of independence are also compared for sample sizes n = 20, 30, 50, empirically. Finally, we apply two examples to illustrate the results.

Keywords

Acknowledgement

Supported by : Ferdowsi University of Mashhad

References

  1. Amini M, Jabbari H, Mohtashami Borzadaran GR, and Azadbakhsh M (2010). Power comparison of independence test for the Farlie-Gumbel-Morgenstern family, Communications of the Korean Statistical Society, 17, 493-505.
  2. Amini M, Jabbari H, and Mohtashami Borzadaran GR (2011). Aspects of dependence in generalized Farlie-Gumbel-Morgenstern distributions, Communications in Statistics - Simulation and Computation, 40, 1192-1205. https://doi.org/10.1080/03610918.2011.568149
  3. Bairamov I and Kotz S (2002). Dependence structure and symmetry of Huang-Kotz FGM distributions and their extensions, Metrika, 56, 55-72. https://doi.org/10.1007/s001840100158
  4. Baker R (2008). An order-statistics-based method for constructing multivariate distributions with fixed marginals, Journal of Multivariate Analysis, 99, 2312-2327. https://doi.org/10.1016/j.jmva.2008.02.019
  5. Cook RD and Johnson ME (1986). Generalized Burr-Pareto-logistic distributions with applications to a uranium exploration data set, Technometrics, 28, 123-131. https://doi.org/10.1080/00401706.1986.10488113
  6. Deheuvels P (1981). An asymptotic decomposition for multivariate distribution-free tests of independence, Journal of Multivariate Analysis, 11, 102-113. https://doi.org/10.1016/0047-259X(81)90136-6
  7. Dou X, Kuriki S, Lin GD, and Richards D (2016). EM algorithms for estimating the Bernstein copula, Computational Statistics & Data Analysis, 93, 228-245. https://doi.org/10.1016/j.csda.2014.01.009
  8. Farlie DJG (1960). The performance of some correlation coefficients for a general bivariate distribution, Biometrika, 47, 307-323. https://doi.org/10.1093/biomet/47.3-4.307
  9. Fasano G and Franceschini A (1987). A multidimensional version of the Kolmogorov-Smirnov test, Monthly Notices of the Royal Astronomical Society, 225, 155-170. https://doi.org/10.1093/mnras/225.1.155
  10. Genest C, Quessy JF, and Remillard B (2006). Local efficiency of a Cramer-von Mises test of independence, Journal of Multivariate Analysis, 97, 274-294. https://doi.org/10.1016/j.jmva.2005.03.003
  11. Genest C and Remillard B (2004). Test of independence and randomness based on the empirical copula process, Test, 13, 335-369. https://doi.org/10.1007/BF02595777
  12. Genest C and Verret F (2005). Locally most powerful rank tests of independence for copula models, Journal of Nonparametric Statistics, 17, 521-539. https://doi.org/10.1080/10485250500038926
  13. Gumbel EJ (1960). Bivariate exponential distributions, Journal of the American Statistical Association, 55, 698-707. https://doi.org/10.1080/01621459.1960.10483368
  14. Guven B and Kotz S (2008). Test of independence for generalized Farlie-Gumbel-Morgenstern distributions, Journal of Computational and Applied Mathematics, 212, 102-111. https://doi.org/10.1016/j.cam.2006.11.029
  15. Hajek J and Sidak Z (1967). Theory of Rank Tests, Academic Press, San Diego, CA.
  16. Hlubinka D and Kotz S (2010). The generalized FGM distribution and its application to stereology of extremes, Applications of Mathematics, 55, 495-512. https://doi.org/10.1007/s10492-010-0020-x
  17. Huang JS and Kotz S (1999). Modifications the Farlie-Gumbel-Morgenstern distributions: a tough hill to climb, Metrika, 49, 135-145. https://doi.org/10.1007/s001840050030
  18. Jung YS, Kim JM, and Kim J (2008). New approach of directional dependence in exchange markets using generalized FGM copula function, Communications in Statistics - Simulation and Computation, 37, 772-788. https://doi.org/10.1080/03610910701711091
  19. Kendall MG and Gibbons JD (1990). Rank Correlation Methods (5th ed), Oxford University Press, New York.
  20. Kochar SC and Gupta RP (1987). Competitors of Kendall-tau test for testing independence against positive quadrant dependence, Biometrika, 74, 664-666. https://doi.org/10.1093/biomet/74.3.664
  21. Kochar SC and Gupta RP (1990). Distribution-free tests based on sub-sample extrema for testing against positive dependence, Australian Journal of Statistics, 32, 45-51. https://doi.org/10.1111/j.1467-842X.1990.tb00998.x
  22. Lehmann EL (1966). Some concepts of dependence, Annals of Mathematics and Statistics, 37, 1137-1153. https://doi.org/10.1214/aoms/1177699260
  23. Morgenstern D (1956). Einfache beispiele zweidimensionaler verteilungen, Mitteilungsblatt fur Mathematische Statistik, 8, 234-235.
  24. Nelsen RB (2006). An Introduction to Copulas (2nd ed), Springer, New York.
  25. Peacock JA (1983). Two-dimensional goodness-of-fit testing in astronomy, Monthly Notices of the Royal Astronomical Society, 202, 615-627. https://doi.org/10.1093/mnras/202.3.615
  26. Rodel E and Kossler W (2004). Linear rank tests for independence in bivariate distributions: power comparisons by simulation, Computational Statistics & Data Analysis, 46, 645-660. https://doi.org/10.1016/j.csda.2003.09.005
  27. Serfling RJ (1980). Approximations Theorems of Mathematical Statistics, John Wiley & Sons, New York.
  28. Shetty ID and Pandit PV (2003). Distribution-free tests for independence against positive quadrant dependence: a generalization, Statistical Methods and Applications, 12, 5-17. https://doi.org/10.1007/BF02511580
  29. 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.