A Comparison of Copula Families on Dependence Structure of Extreme Order Statistics

Abstract

Order statistics have an important place in probability theory and statistical inference, especially reliability analysis. It is aimed to compare the dependency structure with some copula families of min-max copula for the extreme order statistics in this study. Suitability of Clayton, Frank, Gumbel from Archimedean copula families, Gaussian, Farlie-Gumbel-Morgenstern and       min-max copula are examined by simulation study. To find the most suitable copula family, some model selection criteries are used and important inferences are obtained.

Keywords

• Dependent structure,order statistics,copula,simulation

References

1. [1] Li X., Fang R., “Ordering properties of order statistics from random variables of Archimedean copulas with applications”, Journal of Multivariate Analysis, 133:304-320, (2015).
2. [2] Arnold B.C., Balakrishnan N., Nagaraja H.N., “A First Course in Order Statistics”, Vol. 54, John Wiley&Sons, New York, (1992).
3. [3] Balakrishnan N., Rao C. R., “Handbook of Statistics 16: Order Statistics: Theory and Methods”, Elsevier, New York, (1998a).
4. [4] Balakrishnan N., Rao C. R., “Handbook of Statistics 17: Order Statistics: Applications”, Vol. 42, Elsevier, New York, (1998b).
5. [5] David H. A., Nagaraja H. N., “Order Statistics”, Wiley&Sons, Third edition, New Jersey, (2003).
6. [6] Schmitz V., “Revealing the dependence structure between and ”, Journal of Statistical Planning Inference, 123(1):41–47, (2004).
7. [7] Li X., Li Z., “Proof of a conjecture on Spearman’s and Kendall’s for sample minimum and maximum”, Journal of Statistical Planning Inference, 137(1):359–361, (2007).
8. [8] Chen YP., “A note on the relationship between Spearman’s and Kendall’s for extreme order statistics”, Journal of Statistical Planning Inference, 137(7):2165–2171, (2007).

9. [9] Ghalibaf M. B., “Dependent structure of extreme order statistics”, Journal of Statistical Computation and Simulation, 86:2846-2855, (2016).
10. [10] Nelsen R.B., “An Introduction to Copulas, in: Lecture Notes in Statistics” Vol. 139, Springer,New York, (1999).
11. [11] Embrechts P., Lindskog F., McNeil A., “Modeling dependence with copulas and applications to risk management”, In: S.T. Rachev, editor, Handbook of Heavy Tailed Distribution in Finance, JAI Press: Handbooks in Finance, 357-360, (2003).
12. [12] Genest C., Favre A.C., “Everything you always wanted to know about copula modeling but were afraid to ask”, Journal of Hydrologic Engineering,12(4):347-368, (2007).
13. [13] Li D.Q., Tang X.S., Phoon K.K. and Chen Y.F., “Bivariate simulation using copula and its application to probabilistic pile settlement analysis”, International Journal Numerical Analytical Methods Geomechanics, 37:597-617, (2013).
14. [14] Genest C., Rivest L. P.,” Statistical inference procedures for bivariate Archimedean copulas”, Journal of the American Statistical Association, 88 (3): 1034-1043 (1993).
15. [15] Akaike H., “A new look at the statistical model identification”, IEEE Transactions on Automatic Control, 19(6), 716-723, (1974).
16. [16] Schwarz G., “Estimating the dimension of a model”, The annals of Statistics, 6(2), 461-464, (1978).