Functional equations involving Sibuya’s dependence function
We introduce a new probability aging notion via a functional equation based on the tail invariance of Sibuya’s dependence function which is specified as the ratio between the joint survival function and the product of its marginal survival functions. Solutions of the functional equation are generated by Gumbel’s type I bivariate exponential distribution and independence law. In a particular setting, we construct a version of Gumbel’s law with a singular component.
KeywordsBivariate lack of memory property Characterization Copula Dependence function Functional equations Gumbel’s type I bivariate exponential distribution Hazard rate Marshall–Olkin’s fatal shock model
Mathematics Subject ClassificationPrimary 60E05 62N05 Secondary 60K10
Unable to display preview. Download preview PDF.
The authors are grateful for the referees suggestions which highly improved the earlier version of the article. The first author is partially supported by FAPESP Grant No. 2013/07375-0.
- 2.Charpentier, A.: Tail distribution and dependence measure. In: Proceedings of the 34th ASTIN Conference (2003)Google Scholar
- 3.Charpentier, A.: Dependence Structure and Limiting Results with Applications in Finance and Insurance. Ph.D. thesis, Katholieke Universiteit of Leuven (2006)Google Scholar
- 14.Pinto, J., Kolev, N.: Extended Marshall–Olkin model and its dual version. In: Cherubini, U., Durante, F., Mulinacci, S. (eds.) Marshall–Olkin Distributions—Advances in Theory and Applications. Springer Series in Probability and Statistics 141, Chapter 6, 87–113 (2015b)Google Scholar
- 15.Pinto, J., Kolev, N.: Copula representations for invariant dependence functions. In: Glau, K., Scherer, M., Zagst, R. (eds) Innovations in Quantitative Risk Management. Springer Series in Probability and Statistics 99, Chapter 24, 411–421 (2015c)Google Scholar