Aequationes mathematicae

, Volume 92, Issue 3, pp 441–451 | Cite as

Functional equations involving Sibuya’s dependence function

  • Nikolai Kolev
  • Jayme Pinto


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.


Bivariate 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 Classification

Primary 60E05 62N05 Secondary 60K10 


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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.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of São PauloSão PauloBrazil

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