Generalized madogram and pairwise dependence of maxima over two regions of a random field

summary:Spatial environmental processes often exhibit dependence in their large values. In order to model such processes their dependence properties must be characterized and quantified. In this paper we introduce a measure that evaluates the dependence among extreme observations located in two disjoint sets of locations of $\mathbb{R}^2$. We compute the range of this new dependence measure, which extends the existing $\lambda$-madogram concept, and compare it with extremal coefficients, finding generalizations of the known relations in the pairwise approach. Estimators for this measure are introduced and asymptotic normality and strong consistency are shown. An application to the annual maxima precipitation in Portuguese regions is presented

Spatial modeling of extreme snow depth

The spatial modeling of extreme snow is important for adequate risk management in Alpine and high altitude countries. A natural approach to such modeling is through the theory of max-stable processes, an infinite-dimensional extension of multivariate extreme value theory. In this paper we describe the application of such processes in modeling the spatial dependence of extreme snow depth in Switzerland, based on data for the winters 1966--2008 at 101 stations. The models we propose rely on a climate transformation that allows us to account for the presence of climate regions and for directional effects, resulting from synoptic weather patterns. Estimation is performed through pairwise likelihood inference and the models are compared using penalized likelihood criteria. The max-stable models provide a much better fit to the joint behavior of the extremes than do independence or full dependence models.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS464 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Approximate Bayesian computing for spatial extremes

Statistical analysis of max-stable processes used to model spatial extremes has been limited by the difficulty in calculating the joint likelihood function. This precludes all standard likelihood-based approaches, including Bayesian approaches. In this paper we present a Bayesian approach through the use of approximate Bayesian computing. This circumvents the need for a joint likelihood function by instead relying on simulations from the (unavailable) likelihood. This method is compared with an alternative approach based on the composite likelihood. When estimating the spatial dependence of extremes, we demonstrate that approximate Bayesian computing can provide estimates with a lower mean square error than the composite likelihood approach, though at an appreciably higher computational cost. We also illustrate the performance of the method with an application to US temperature data to estimate the risk of crop loss due to an unlikely freeze event

Multivariate Nonparametric Estimation of the Pickands Dependence Function using Bernstein Polynomials

Many applications in risk analysis, especially in environmental sciences, require the estimation of the dependence among multivariate maxima. A way to do this is by inferring the Pickands dependence function of the underlying extreme-value copula. A nonparametric estimator is constructed as the sample equivalent of a multivariate extension of the madogram. Shape constraints on the family of Pickands dependence functions are taken into account by means of a representation in terms of a specific type of Bernstein polynomials. The large-sample theory of the estimator is developed and its finite-sample performance is evaluated with a simulation study. The approach is illustrated by analyzing clusters consisting of seven weather stations that have recorded weekly maxima of hourly rainfall in France from 1993 to 2011

Statistical Modeling of Spatial Extremes

The areal modeling of the extremes of a natural process such as rainfall or temperature is important in environmental statistics; for example, understanding extreme areal rainfall is crucial in flood protection. This article reviews recent progress in the statistical modeling of spatial extremes, starting with sketches of the necessary elements of extreme value statistics and geostatistics. The main types of statistical models thus far proposed, based on latent variables, on copulas and on spatial max-stable processes, are described and then are compared by application to a data set on rainfall in Switzerland. Whereas latent variable modeling allows a better fit to marginal distributions, it fits the joint distributions of extremes poorly, so appropriately-chosen copula or max-stable models seem essential for successful spatial modeling of extremes.Comment: Published in at http://dx.doi.org/10.1214/11-STS376 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org