Spatial hierarchical modeling of threshold exceedances using rate mixtures

TL;DR

This paper introduces a flexible spatial hierarchical model for threshold exceedances that captures tail behavior and spatial dependence, using Bayesian methods and MCMC for efficient inference, demonstrated on rainfall data.

Contribution

The paper develops a novel gamma-gamma hierarchical model for spatial threshold exceedances that accommodates departures from asymptotic stability and jointly models tails and bulk data.

Findings

Model effectively captures tail and bulk behavior.

Bayesian inference with MALA is computationally efficient.

Application to rainfall data shows good predictive performance.

Abstract

We develop new flexible univariate models for light-tailed and heavy-tailed data, which extend a hierarchical representation of the generalized Pareto (GP) limit for threshold exceedances. These models can accommodate departure from asymptotic threshold stability in finite samples while keeping the asymptotic GP distribution as a special (or boundary) case and can capture the tails and the bulk jointly without losing much flexibility. Spatial dependence is modeled through a latent process, while the data are assumed to be conditionally independent. Focusing on a gamma-gamma model construction, we design penalized complexity priors for crucial model parameters, shrinking our proposed spatial Bayesian hierarchical model toward a simpler reference whose marginal distributions are GP with moderately heavy tails. Our model can be fitted in fairly high dimensions using Markov chain Monte Carlo by exploiting the Metropolis-adjusted Langevin algorithm (MALA), which guarantees fast convergence of Markov chains with efficient block proposals for the latent variables. We also develop an adaptive scheme to calibrate the MALA tuning parameters. Moreover, our model avoids the expensive numerical evaluations of multifold integrals in censored likelihood expressions. We demonstrate our new methodology by simulation and application to a dataset of extreme rainfall events that occurred in Germany. Our fitted gamma-gamma model provides a satisfactory performance and can be successfully used to predict rainfall extremes at unobserved locations.