A Vecchia approximation for high-dimensional Gaussian cumulative distribution functions arising from spatial data

We introduce an approach to quickly and accurately approximate the cumulative distribution function of multivariate Gaussian distributions arising from spatial Gaussian processes. This approximation is trivially parallelizable and simple to implement using standard software. We demonstrate its accuracy and computational efficiency in a series of simulation experiments and apply it to analyzing the joint tail of a large precipitation dataset using a recently-proposed scale mixture model for spatial extremes. This dataset is many times larger than what was previously considered possible to fit using preferred inferential techniques.