Sampling Exchangeable and Hierarchical Marshall-Olkin Distributions

Abstract

The seminal Marshall-Olkin distribution is given by the following d-dimensional frailty model: independent exponentially distributed random variables represent the arrival times of shocks that destroy subgroups of a vector with d components. Sampling the resulting random vector of extinction times along the lines of this stochastic representation is conceptually straightforward. Firstly, all possible shocks are sampled. Secondly, the extinction time of each component is computed. However, implementing this sampling scheme involves 2 d − 1 random variables and requires finding the minimum of a set with 2 d−1 elements for each component. Thus, the overall effort is exponentially increasing in the number of considered components, preventing the general model from being used in high-dimensional simulation studies. This problem is overcome for the subfamily of exchangeable Marshall-Olkin distributions, for which a sampling algorithm with polynomial effort in d, an upper bound is 𝒪(d 3), is presented. In a second step, it is shown how hierarchical models are constructed from exchangeable structures and how models with alternative univariate marginal laws are obtained. The presented algorithm is then adapted to sample such structures as well. Finally, a small case study in the context of portfolio credit risk is presented.