High-dimensional inference using the extremal skew-$t$ process

TL;DR

This paper develops theoretical tools and computational methods for high-dimensional inference of the extremal skew-$t$ process, a flexible model for environmental extremes, enabling efficient analysis of large datasets.

Contribution

It introduces new formulae for simulation and inference of the extremal skew-$t$ process, incorporating Stephenson-Tawn likelihoods into composite likelihood frameworks for high-dimensional data.

Findings

Efficient algorithms for simulating the extremal skew-$t$ process.

Enhanced composite likelihood methods with Stephenson-Tawn integration.

Successful application to a 90-dimensional temperature dataset.

Abstract

Max-stable processes are a popular tool for the study of environmental extremes, and the extremal skew-$t$ process is a general model that allows for a flexible extremal dependence structure. For inference on max-stable processes with high-dimensional data, exact likelihood-based estimation is computationally intractable. Composite likelihoods, using lower dimensional components, and Stephenson-Tawn likelihoods, using occurrence times of maxima, are both attractive methods to circumvent this issue for moderate dimensions. In this article we establish the theoretical formulae for simulations of and inference for the extremal skew-$t$ process. We also incorporate the Stephenson-Tawn concept into the composite likelihood framework, giving greater statistical and computational efficiency for higher-order composite likelihoods. We compare 2-way (pairwise), 3-way (triplewise), 4-way, 5-way and 10-way composite likelihoods for models of up to 100 dimensions. Furthermore, we propose cdf approximations for the Stephenson-Tawn likelihood function, leading to large computational gains, and enabling accurate fitting of models in large dimensions in only a few minutes. We illustrate our methodology with an application to a 90-dimensional temperature dataset from Melbourne, Australia.