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An urn approach to generalized extreme shock models

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  • Cirillo, Pasquale
  • Hüsler, Jürg

Abstract

We introduce a new intuitive approach to generalized extreme shock models (GESM) using urn processes. This allows us to indirectly model the moving risky threshold of generalized extreme shock models introduced in [Gut, A., Hsler, J., 2005. Realistic variation of shock models. Statistics & Probability Letters 74, 187-204]. The basic idea is to link the colors of the balls in the urn with the levels of risk a system can face, and model the evolution of the process using a triangular reinforcement matrix. Using the analytic approach proposed in [Flajolet, P., Gabarro, J., Pekari, H., 2005. Analytic urns. Annals of Probability 33, 1200-1233], we explicitly derive probabilities, moments and limit laws for different types of balls, that is for the distinct levels of risk. Our model can be also considered as a way to incorporate GESM in a Bayesian framework. In fact, urn processes are an important tool of Bayesian Nonparametrics, in the way they allow us to compute posterior distributions without an explicit knowledge of priors.

Suggested Citation

  • Cirillo, Pasquale & Hüsler, Jürg, 2009. "An urn approach to generalized extreme shock models," Statistics & Probability Letters, Elsevier, vol. 79(7), pages 969-976, April.
    • Handle: RePEc:eee:stapro:v:79:y:2009:i:7:p:969-976
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    References listed on IDEAS

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    1. Gut, Allan & Hüsler, Jürg, 2005. "Realistic variation of shock models," Statistics & Probability Letters, Elsevier, vol. 74(2), pages 187-204, September.
    2. R. Pillai, 1990. "On Mittag-Leffler functions and related distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(1), pages 157-161, March.
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    Cited by:

    1. Pasquale Cirillo & Jürg Hüsler & Pietro Muliere, 2013. "Alarm Systems and Catastrophes from a Diverse Point of View," Methodology and Computing in Applied Probability, Springer, vol. 15(4), pages 821-839, December.
    2. Janßen Anja & Segers Johan, 2024. "Invariance properties of limiting point processes and applications to clusters of extremes," Dependence Modeling, De Gruyter, vol. 12(1), pages 1-12, January.
    3. Cirillo, Pasquale & Hüsler, Jürg, 2011. "Extreme shock models: An alternative perspective," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 25-30, January.

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