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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
CreditRisk+; R
Full Text:
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