An urn approach to generalized extreme shock models

Abstract

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., Husler, 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.