Archimedean copulae and positive dependence. (English) Zbl 1065.60018
Summary: We consider different issues related to Archimedean copulae and positive dependence. In the first part, we characterize Archimedean copulae that possess positive dependence properties such as multivariate total positivity of order 2 (MTP\(_2\)) and conditionally increasingness in sequence. In the second part, we investigate conditions for exchangeable binary sequences to admit an Archimedean copula, and we show that they depend on the extendibility of the sequence, and therefore on its positive-dependence properties.
MSC:
| 60E15 | Inequalities; stochastic orderings |
| 62G09 | Nonparametric statistical resampling methods |
| 62H10 | Multivariate distribution of statistics |
| 62H20 | Measures of association (correlation, canonical correlation, etc.) |
Keywords:
Conditionally increasing; MTP\(_2\); Positive lower orthant dependent; Exchangeability; Binary sequencesReferences:
| [1] | Averous, J.; Dortet-Bernadet, J.-L., LTD and RTI dependence orderings, Canad. J. Statist., 28, 1, 151-157 (2000) · Zbl 1066.62527 |
| [2] | Bagdonavičius, V.; Malov, S.; Nikulin, M., Characterizations and semiparametric regression estimation in Archimedean copulas, J. Appl. Statist. Sci., 8, 137-153 (1999) · Zbl 0923.62052 |
| [3] | Ballerini, R., Archimedean copulas, exchangeability, and max-stability, J. Appl. Probab., 31, 383-390 (1994) · Zbl 0812.60022 |
| [4] | Barlow, R. E.; Proschan, F., Statistical Theory of Reliability and Life Testing (1975), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York · Zbl 0379.62080 |
| [5] | B. Bassan, F. Spizzichino, Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes, Mimeo, 2002.; B. Bassan, F. Spizzichino, Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes, Mimeo, 2002. · Zbl 1070.60015 |
| [6] | Capéraà, P.; Genest, C., Concepts de dépendance et ordres stochastiques pour des lois bidimensionnelles, Canad. J. Statist., 18, 315-326 (1990) · Zbl 0738.62057 |
| [7] | Capéraà, P.; Genest, C., Spearman’s \(ρ\) is larger than Kendall’s \(τ\) for positively dependent random variables, J. Nonparametr. Statist., 2, 183-194 (1993) · Zbl 1360.62294 |
| [8] | Cuculescu, L.; Theodorescu, R., Are copulas unimodal?, J. Multivariate Anal., 86, 48-71 (2003) · Zbl 1028.60009 |
| [9] | Deheuvels, P., Caractérisation complète des lois extrêmes multivariées et de la convergence des types extrêmes, Publ. Inst. Statist. Univ. Paris, 23, 1-37 (1978) · Zbl 0414.60043 |
| [10] | M. Denuit, C. Genest, É. Marceau, Criteria for the stochastic ordering of random sums, with actuarial applications, Scand. Actuar. J. (2002) 3-16.; M. Denuit, C. Genest, É. Marceau, Criteria for the stochastic ordering of random sums, with actuarial applications, Scand. Actuar. J. (2002) 3-16. · Zbl 1003.60022 |
| [11] | Embrechts, P.; Höing, A.; Juri, A., Using copulae to bound the value-at-risk for functions of dependent risks, Finance Stochastics, 7, 145-167 (2003) · Zbl 1039.91023 |
| [12] | De Finetti, B., Theory of Probability (1990), Wiley: Wiley Chichester · Zbl 0694.60001 |
| [13] | Frees, E. W.; Valdez, E., Understanding relationships using copulas, North Amer. Actuar. J., 2, 1-25 (1998) · Zbl 1081.62564 |
| [14] | R. Frey, A.J. McNeil, Modelling dependent defaults, Mimeo, 2001.; R. Frey, A.J. McNeil, Modelling dependent defaults, Mimeo, 2001. |
| [15] | Genest, C.; MacKay, R. J., Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données, Canad. J. Statist., 14, 145-159 (1986) · Zbl 0605.62049 |
| [16] | Genest, C.; Marceau, É.; Mesfioui, M., Compound Poisson approximations for individual models with dependent risks, Ins. Math. Econ., 32, 73-91 (2003) · Zbl 1055.91050 |
| [17] | Genest, C.; Rivest, L.-P., Statistical inference procedures for bivariate Archimedean copulas, J. Amer. Statist. Assoc., 88, 1034-1043 (1993) · Zbl 0785.62032 |
| [18] | Hennessy, D. A.; Lapan, H. E., The use of Archimedean copulas to model portfolio allocations, Math. Finance, 12, 143-154 (2002) · Zbl 1072.91022 |
| [19] | Joe, H., Multivariate Models and Dependence Concepts (1997), Chapman & Hall: Chapman & Hall London · Zbl 0990.62517 |
| [20] | Juri, A.; Wüthrich, A. V., Copula convergence theorems for tail events, Ins. Math. Econom., 30, 405-420 (2002) · Zbl 1039.62043 |
| [21] | Karlin, S.; Rinott, Y., Classes of orderings of measures and related correlation inequalities, I. Multivariate totally positive distributions, J. Multivariate Anal., 10, 467-498 (1980) · Zbl 0469.60006 |
| [22] | Kimberling, C. H., A probabilistic interpretation of complete monotonicity, Aequationes Math., 10, 152-164 (1974) · Zbl 0309.60012 |
| [23] | Kimeldorf, G.; Sampson, A., Uniform representations of bivariate distributions, Comm. Statist., 4, 617-627 (1975) · Zbl 0312.62008 |
| [24] | Lehmann, E. L., Some concepts of dependence, Ann. Math. Statist., 37, 1137-1153 (1966) · Zbl 0146.40601 |
| [25] | Marshall, A. W.; Olkin, I., Families of multivariate distributions, J. Amer. Statist. Assoc., 83, 834-841 (1988) · Zbl 0683.62029 |
| [26] | Müller, A.; Scarsini, M., Stochastic comparison of random vectors with a common copula, Math. Oper. Res., 26, 723-740 (2001) · Zbl 1082.60504 |
| [27] | Müller, A.; Stoyan, D., Comparison Methods for Stochastic Models and Risks (2002), Wiley Series in Probability and Statistics, Wiley: Wiley Series in Probability and Statistics, Wiley Chichester · Zbl 0999.60002 |
| [28] | Nelsen, R. B., An Introduction to Copulas (1999), Springer: Springer New York · Zbl 0909.62052 |
| [29] | Oakes, D., Multivariate survival distributions, J. Nonparametr. Statist., 3, 343-354 (1994) · Zbl 1378.62121 |
| [30] | Roberts, A. W.; Varberg, D. E., Convex Functions (1973), Academic Press: Academic Press New York-London · Zbl 0289.26012 |
| [31] | T. Roncalli, Gestion des risques multiples, Technical report, Groupe de Recherche Opérationelle, Crédit Lyonnais, 2001.; T. Roncalli, Gestion des risques multiples, Technical report, Groupe de Recherche Opérationelle, Crédit Lyonnais, 2001. |
| [32] | Schweizer, B.; Sklar, A., Probabilistic Metric Spaces (1983), North-Holland: North-Holland New York · Zbl 0546.60010 |
| [33] | Sklar, M., Fonctions de répartition à \(n\) dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, 8, 229-231 (1959) · Zbl 0100.14202 |
| [34] | Wei, G.; Hu, T., Supermodular dependence ordering on a class of multivariate copulas, Statist. Probab. Lett., 57, 375-385 (2002) · Zbl 1005.60037 |
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