Spatial hierarchical modeling of threshold exceedances using rate mixtures

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.

A parametric model for distributions with flexible behavior in both tails

“…For the family , it might correspondingly be appropriate to assume does not vary over some region or, if that did not suffice to produce stable parameter estimates, to assume , and do not vary. More recent work on environmental extremes often uses hierarchical models to force the scale and shape parameters of the distribution to vary smoothly in space (Bacro, Gaetan, Opitz, & Toulemonde, 2020; Castro‐Camilo, Huser, & Rue, 2019; Sharkey & Winter, 2019; Yadav et al, 2019) and this approach could also be applied to the parametric families proposed here. Such analyses are usually Bayesian and it is not clear what kinds of priors should be put on the parameters as they vary in some spatial index x .…”

“…All of these approaches require numerical computation of the normalizing constant for the density. Yadav, Huser, and Opitz (2019) use mixtures of powers of gamma random variables to model distributions of positive random variables whose upper tail index is nonnegative. Much closer to the work here, Papastathopoulos and Tawn (2013) and Naveau, Huser, Ribereau, and Hannart (2016) compose two cumulative distribution functions to describe models for positive quantities.…”

A flexible Bayesian hierarchical modeling framework for spatially dependent peaks-over-threshold data

“…As an alternative solution, in this work we extend the hierarchical modeling framework developed by Yadav et al (2021) to capture relatively strong spatial dependence among threshold exceedances, by mixing several random fields multiplicatively in a way that ensures a heavy-tailed marginal behavior and generates a wide range of joint tail structures. To propose flexible heavytailed models, Yadav et al (2021) relied on Breiman's Lemma (Breiman, 1965), which characterizes the tail behavior of the product of two nonnegative independent random variables when one of them has power-law tail decay. Let X 1 and X 2 be nonnegative independent random variables such that E(X α+ 1 ) < ∞, for some > 0, and the distribution of X 2 is regularly varying at ∞ with index −α < 0, i.e., Pr (X 2 > x) = (x)x −α , where (x) > 0 and (tx)/ (t) → 1, as t → ∞.…”

“…Breiman's Lemma (1) motivates the construction of models with improved flexibility at a sub-asymptotic level, while allowing a heavytailed behavior. Yadav et al (2021) used this result to generalize the GP distribution, which can be obtained from the product of two independent Exponential and Inverse Gamma-distributed random variables. Specifically, they proposed spatial models constructed as Y…”

“…The conditional independence assumption at the data-level here strongly restricts the form of dependence of Y (s). In this paper, we generalize the hierarchical spatial model of Yadav et al (2021) to provide new, more flexible hierarchical spatial models for threshold exceedances, that mitigate the effect of the conditional independence assumption at the data-level with the ultimate goal of capturing stronger spatial dependence among extreme events, while retaining the computational benefits and the intuitive interpretation of such Bayesian hierarchical models.…”