UCSC-SOE-22-02: Bayesian non-parametric inference for multivariate peaks over threshold models

Peter Trubey and Bruno Sanso
01/14/2022 10:40 AM
We consider a constructive definition of the multivariate Pareto that factorizes the random vector into a radial component and an independent angular component; The former following a univariate Pareto distribution, and the latter defined on the surface of the positive orthant of the unit hypercube. In this paper, we propose a method for inferring the distribution of the angular component. We identify its support as the limit of the positive orthants of the unit p–norm spheres. To this effect, we introduce a pro- jected gamma family of distributions defined as the projection of a vector of independent gamma random variables onto the p–norm sphere. This family serves as a building block for a flexible family of distributions obtained as a Dirichlet process mixture of projected gammas. For model assessment and comparison, we discuss model scoring methods ap- propriate to distributions on the unit hypercube. In particular, working with the energy score criterion, we develop a kernel metric appropriate to the hypercube that produces a proper scoring rule. We then present a simulation study to compare different modeling choices using the proposed scoring rules. Finally we apply our approach to describe the dependence structure of the extreme values of the magnitude of the integrated vapor transport (IVT), a variable that describes the rate of flow of moisture in the atmosphere along the coast of California for the years of 1979 through 2020. We find a clear but heterogeneous geographical dependence.