Multivariate archimedean copulas, d-monotone functions and l 1-norm symmetric distributions

Annals of Statistics

Volumn 37, Issue 5 B, 2009, Pages 3059-3097

  McNeil, Alexander J ^a   Nešlehová, Johanna ^b  

^a HERIOT WATT UNIVERSITY   (United Kingdom)

^b ETH ZURICH   (Switzerland)

Author keywords

Archimedean copula; D monotone function; Dependence ordering; Frailty model; J norm symmetric distribution; Laplace transform; Stochastic simulation; Williamson d transform

Indexed keywords

EID: 68049091067     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/07-AOS556     Document Type: Article

References (41)

1. Barbe, P., Genest, C., Ghoudi, K. and Remillard, B. (1996). On Kendall's process. J. Multivariate Anal. 58 197-229. MR1405589
2. Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Encyclopedia of Mathematics and its Applications 27. Cambridge Univ. Press, Cambridge. MR1015093
3. Boas, Jr., R. P. and Widder, D. V. (1940). Functions with positive differences. Duke Math. J. 7 496-503. MR0003436
4. Cherubini, U., Luciano, E.andVecchiato, W. (2004). Copula Methods in Finance. Wiley, Chichester. MR2250804
5. Clayton, D. G. (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65 141-151. MR0501698
6. Easton, R.J., Tucker, D. H. and Wayment, S. G. (1967). On the existence almost everywhere of the cross partial derivatives. Math. Z. 102 171-176. MR0218502
7. Fang, K.-T. and Fang, B.-Q. (1988). Some families of multivariate symmetric distributions related to exponential distribution. J. Multivariate Anal. 24 109-122. MR0925133
8. Fang, K.-T., Kotz, S. and Ng, K.-W. (1990). Symmetric Multivariate and Related Distributions. Chapman and Hall, London. MR1071174
9. Frees, E.W. and Valdez, E. A. (1997). Understanding relationships using copulas. North American Actuarial J. 2 1-25. MR1988432
10. Genest, C., Ghoudi, K. and Rivest, L.-P. (1995). A semi-parametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82 543-552. MR1366280
11. Genest, C. and MacKay, R. J. (1986a). Copules archimédiennes et familles de lois bidi- mensionnelles dont les marges sont donnees. Canad. J. Statist. 14 145-159. MR0849869
12. Genest, C. and MacKay, R. J. (1986b). The joy of copulas: Bivariate distributions with uniform marginals. Amer. Statist. 40 280-283. MR0866908
13. Genest, C. and Neslehová, J. (2007). A primer on copulas for discrete data. Astin Bull. 37 475-515.
14. Genest, C. and Rivest, L.-P. (1989). A characterization of Gumbel's family of extreme value distributions. Statist. Probab. Lett. 8 207-211. MR1024029
15. Genest, C. and Rivest, L.-P. (1993). Statistical inference procedures for bivariate Archimedean copulas. J. Amer. Statist. Assoc. 88 1034-1043. MR1242947
16. Hewitt, E. and Stromberg, K. (1975). Real and Analysis. Springer, New York. MR0367121
17. Hoffmann-JøRGENSEN, J. (1994). Probability with a View Toward Statistics. II. Chapman and Hall, New York.
18. Hopf, E. (1926). Uber die Zusammenhänge zwischen gewissen höheren Differenzenquotienten reeller Funktionen einer reellen Variablen und deren Differenzierbarkeitseigenschaften. Ph.D. thesis, Univ. Berlin.
19. Joe, H. (1997). Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability 73. Chapman and Hall, London. MR1462613
20. Kimberling, C. H. (1974). A probabilistic interpretation of complete monotonicity. Aequa- tionesMath. 10 152-164. MR0353416
21. Klugman, S. A. and Parsa, R. (1999). Fitting bivariate loss distributions with copulas. Insurance Math. Econom. 24 139-148. MR1710816
22. Ling, C.-H. (1965). Representation of associative functions. Publ. Math. Debrecen 12 189-212. MR190575
23. Marshall, A. W. and Olkin, I. (1988). Families of multivariate distributions. J. Amer. Statist. Assoc. 83 834-841. MR0963813
24. McNeil, A. J. (2008). Sampling nested Archimedean copulas. J. Statist. Comput. Simulation. 78 567-581.
25. Miller, A. D. and Vyborny, R. (1986). Some remarks on functions with one-sided derivatives. Amer. Math. Monthly 93 471-475. MR1540891
26. MÜller, A. and Scarsini, M. (2005). Archimedean copulae and positive dependence. J. Multivariate Anal. 93 434-445. MR2162647
27. Nelsen, R. B. (1999). An Introduction to Copulas. Lecture Notes in Statist. 139. Springer, New York. MR1653203
28. Nelsen, R. B. (2005). Some properties of Schur-constant survival models and their copulas., Braz. J. Probab. Stat. 19 179-190. MR2345487
29. Oakes, D. (1989). Bivariate survival models induced by frailties. J. Amer. Statist. Assoc. 84 487-493. MR1010337
30. Oakes, D. (1994). Multivariate survival distributions. J. Nonparametr. Statist. 3 343-354. MR1291555
31. Popoviciu, T. (1933). Sur quelques proprietes des fonctions d'une ou de deux variables reelles. Ph.D. thesis, Faculte des Sciences de Paris.
32. Popoviciu, T. (1944). Les fonctions convexes. Hermann, Paris. MR0018705
33. Roberts, A. W. and Varberg, D. E. (1973). Convex Functions. Pure and Applied Mathematics 57. Academic Press, New York-London. MR0442824
34. Saks, S. (1937). Theory of the Integral, 2nd ed. Hafner Publishing Co., New York.
35. Scarsini, M. (1984). On measures of concordance. Stochastica 8 201-218. MR0796650
36. Schönbucher, P. (2003). Credit Derivatives Pricing Models. Wiley, Chichester.
37. Schweizer, B. and Sklar, A. (1983). Probabilistic Metric Spaces. North-Holland Series in Probability and Applied Mathematics. North-Holland Publishing Co., New York. MR0790314
38. Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 229-231. MR0125600
39. Whelan, N. (2004). Sampling from Archimedean copulas. Quant. Finance 4 339-352. MR2071842
40. Widder, D. V. (1946). The Laplace Transform. Princeton Univ. Press, Princeton.
41. Williamson, R. E. (1956). Multiply monotone functions and their Laplace transforms. Duke Math. J. 23 189-207. MR0077581