×

On the construction of nested Archimedean copulas for \(d\)-monotone generators. (English) Zbl 1328.62316

Summary: Following A. J. McNeil and J. Nešlehová [Ann. Stat. 37, No. 5B, 3059–3097 (2009; Zbl 1173.62044)], we present some weaker conditions under which a partially nested Archimedean copula with arbitrary nesting levels is still a copula. Relaxing the conditions on the generators enable researchers to model data sets more efficiently.

MSC:

62H10 Multivariate distribution of statistics
60E05 Probability distributions: general theory
60K10 Applications of renewal theory (reliability, demand theory, etc.)

Citations:

Zbl 1173.62044
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Craik, A. D.D., Prehistory of Faà di Bruno’s formula, Amer. Math. Monthly, 112, 2, 119-130 (2005) · Zbl 1088.01008
[2] Feller, W., An Introduction to Probability Theory and its Applications (1972), John Wiley: John Wiley New York · Zbl 0158.34902
[3] Fermanian, J.-D., Goodness-of-fit tests for copulas, J. Multivariate Anal., 95, 119-152 (2005) · Zbl 1095.62052
[4] Genest, C.; Quessy, J.-F.; Rèmillard, B., Goodness-of-fit procedures for copula models based estimators of the association parameter in the gamma fraily model, Statist. Probab. Lett., 76, 10-18 (2006)
[5] Genest, C.; Rèmillard, B.; Beaudoin, D., Goodness-of-fit tests for copulas: a review and a power study, Insurance Math. Econom., 44, 199-213 (2009), On the probability integral transform. Scandin. J. Statist. 33, 337-366 · Zbl 1161.91416
[6] Hofert, M., A stochastic representation and sampling algorithm for nested Archimedean copulas, J. Stat. Comput. Simul., 82, 9, 1239-1255 (2012) · Zbl 1271.60026
[7] Hofert, M.; Pham, D., Densities of nested Archimedean copulas, J. Multivariate Anal., 118, 37-52 (2013) · Zbl 1277.62138
[8] Joe, H., Multivariate Models and Dependence Concepts (1997), Chapman & Hall: Chapman & Hall London · Zbl 0990.62517
[9] Marshall, A. W.; Olkin, I., Families of multivariate distributions, J. Amer. Statist. Assoc., 83, 834-841 (1988) · Zbl 0683.62029
[10] McNeil, A., Sampling nested Archimedean copulas, J. Stat. Comput. Simul., 78, 567-581 (2008) · Zbl 1221.00061
[11] McNeil, A. J.; Nešlehová, J., Multivariate Archimedean copulas, \(n\)-monotone functions and L1-norm symmetric distributions, Ann. Statist., 37, 3059-3097 (2009) · Zbl 1173.62044
[12] Nelsen, R. B., An Introduction to Copulas (2006), Springer: Springer New York · Zbl 1152.62030
[13] Rezapour, M.; Alamatsaz, M. H., Stochastic comparison of lifetimes of two \((n - k + 1)\)-out-of-\(n\) systems with heterogeneous dependent components, J. Multivariate Anal., 130, 240-251 (2014) · Zbl 1360.62498
[14] Rezapour, M.; Alamatsaz, M. H.; Pellerey, F., Multivariate ageing with Archimedean dependence structures in high dimensions, Comm. Statist. Theory Methods, 42, 2056-2070 (2013) · Zbl 1274.60262
[15] Williamson, R., Multiply monotone functions and their Laplace transforms, Duke Math. J., 23, 189-207 (1956) · Zbl 0070.28501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.