Nonparametric Spatial Models for Extremes: Application to Extreme Temperature

Estimating the probability of extreme temperature events is difficult because of limited records across time and the need to extrapolate the distributions of these events, as opposed to just the mean, to locations where observations are not available. Another related issue is the need to characterize the uncertainty in the estimated probability of extreme events at different locations. Although the tools for statistical modeling of univariate extremes are welldeveloped, extending these tools to model spatial extreme data is an active area of research. In this paper, in order to make inference about spatial extreme events, we introduce two new models. The first one is a Dirichlet-type mixture model, with marginals that have generalized extreme value (GEV) distributions with spatially varying parameters, and the observations are spatially-correlated even after accounting for the spatially varying parameters. This model avoids the matrix inversion needed in the spatial copula frameworks, and it is very computationally efficient. The second is a Dirichlet prior copula model that is a flexible alternative to parametric copula models such as the normal and t-copula. This presents the most flexible multivariate copula approach in the literature. The proposed modelling approaches are fitted using a Bayesian framework that allow us to take into account different sources of uncertainty in the data and models. To characterize the complex dependence structure in the extreme events we introduce nonstationary (space-dependent) extremalcoefficient functions, and threshold-specific extremal functions. We apply our methods to annual maximum temperature values in the east-south-central United States.