Sampling sup‐normalized spectral functions for Brown–Resnick processes

Abstract

Sup‐normalized spectral functions form building blocks of max‐stable and Pareto processes and therefore play an important role in modelling spatial extremes. For one of the most popular examples, the Brown–Resnick process, simulation is not straightforward. In this paper, we generalize two approaches for simulation via Markov chain Monte Carlo methods and rejection sampling by introducing new classes of proposal densities. In both cases, we provide an optimal choice of the proposal density with respect to sampling efficiency. The performance of the procedures is demonstrated in an example.

A Comparative Tour through the Simulation Algorithms for Max-Stable Processes

“…For Brown-Resnick processes or extremal-t processes sampling from V • ∞ is hard and choosing a generic rejection sampling approach for this task is not particularly helpful, which explains the poor performance of the SN approach in our study. Alternatives include MCMC approaches Zhou, 2018, Oesting, Schlather andSchillings, 2019) and for the class of Brown-Resnick processes modified rejection sampling (Oesting, Schlather and Schillings, 2019) or using the ansatz of Ho and Dombry (2019). For other classes of max-stable processes, the SN approach may well be the most efficient way of exact simulation.…”

“…Finally, the simple representation (4.3) of the sum-normalized spectral functions can also be used to simulate the supnormalized spectral functions via rejection sampling (de Fondeville and . Modifications of the last approach to reduce the rejection rate and alternative MCMC procedures have recently been proposed by Oesting, Schlather and Schillings (2019).…”