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pTAS distributions with application to risk management. (English) Zbl 1349.62520
Summary: The family of positive tempered \(\alpha\)-stable (pTAS) or sometimes also tempered one-sided \(\alpha\)-stable distributions dates back to M. C. K. Tweedie [“An index which distinguishes between some important exponential families”, in: Statistics: application and new directions. Proceedings of the conference, Calcutta, India, 1981. Calcutta: Indian Statistical Institute. 579–604 (1984)] and P. Hougaard [Biometrika 73, 387–396 (1986; Zbl 0603.62015)] who discussed it in the context of frailty distribution in life table methods for heterogenous populations. The pTAS family generalizes the well-known gamma distribution and allows for heavier tails depending on the parameter \(\alpha\). Because of this property, pTAS distributions appear to be useful in the context of risk management. Against this background, the contribution of his work is three-fold: Firstly, we summarize the properties of the pTAS family. Secondly, we describe its numerical implementation and illustrate the functions by means of R examples in the Appendix. Thirdly, we empirically demonstrate that this family can be successfully applied in risk management. Concretely, applications to credit and operational risk are given.
MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
60E07 Infinitely divisible distributions; stable distributions
44A10 Laplace transform
91B30 Risk theory, insurance (MSC2010)
91G40 Credit risk
Software:
CreditRisk+; R
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