  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.
