Many environmental processes exhibit weakening spatial dependence as events become more extreme. Well-known limiting models, such as max-stable or generalized Pareto processes, cannot capture this, which can lead to a preference for models that exhibit a property known as asymptotic independence. However, weakening dependence does not automatically imply asymptotic independence, and whether the process is truly asymptotically (in)dependent is usually far from clear. The distinction is key as it can have a large impact upon extrapolation, i.e., the estimated probabilities of events more extreme than those observed. In this work, we present a single spatial model that is able to capture both dependence classes in a parsimonious manner, and with a smooth transition between the two cases. The model covers a wide range of possibilities from asymptotic independence through to complete dependence, and permits weakening dependence of extremes even under asymptotic dependence. Censored likelihood-based inference for the implied copula is feasible in moderate dimensions due to closed-form margins. The model is applied to oceanographic datasets with ambiguous true limiting dependence structure.

Max-stable processes are the natural analogues of the generalized extreme-value distribution for the modelling of extreme events in space and time. Under suitable conditions, these processes are asymptotically justified models for maxima of independent replications of random fields, and they are also suitable for the modelling of joint individual extreme measurements over high thresholds. This paper extends a model of Schlather (2002) to the space-time framework, and shows how a pairwise censored likelihood can be used for consistent estimation under mild mixing conditions. Estimator efficiency is also assessed and the choice of pairs to be included in the pairwise likelihood is discussed based on computations for simple time series models. The ideas are illustrated by an application to hourly precipitation data over Switzerland.

In statistics, extreme events are often defined as excesses above a given large threshold. This definition allows hydrologists and flood planners to apply Extreme-Value Theory (EVT) to their time series of interest. Even in the stationary univariate context, this approach has at least two main drawbacks. First, working with excesses implies that a lot of observations (those below the chosen threshold) are completely disregarded. The range of precipitation is artificially shopped down into two pieces, namely large intensities and the rest, which necessarily imposes different statistical models for each piece. Second, this strategy raises a nontrivial and very practical difficultly: how to choose the optimal threshold which correctly discriminates between low and heavy rainfall intensities. To address these issues, we propose a statistical model in which EVT results apply not only to heavy, but also to low precipitation amounts (zeros excluded). Our model is in compliance with EVT on both ends of the spectrum and allows a smooth transition between the two tails, while keeping a low number of parameters. In terms of inference, we have implemented and tested two classical methods of estimation: likelihood maximization and probability weighed moments. Last but not least, there is no need to choose a threshold to define low and high excesses. The performance and flexibility of this approach are illustrated on simulated and hourly precipitation recorded in Lyon, France.

Statistics of extremes concerns inference for rare events. Often the events have never yet been observed, and their probabilities must therefore be estimated by extrapolation of tail models fitted to available data. Because data concerning the event of interest may be very limited, efficient methods of inference play an important role. This article reviews this domain, emphasizing current research topics. We first sketch the classical theory of extremes for maxima and threshold exceedances of stationary series. We then review multivariate theory, distinguishing asymptotic independence and dependence models, followed by a description of models for spatial and spatiotemporal extreme events. Finally, we discuss inference and describe two applications. Animations illustrate some of the main ideas. 9.1

Gaussian scale mixtures are constructed as Gaussian processes with a random variance. They have non-Gaussian marginals and can exhibit asymptotic dependence unlike Gaussian processes, which are asymptotically independent except in the case of perfect dependence. In this paper, we study in detail the extremal dependence properties of Gaussian scale mixtures and we unify and extend general results on their joint tail decay rates in both asymptotic dependence and independence cases. Motivated by the analysis of spatial extremes, we propose several flexible yet parsimonious parametric copula models that smoothly interpolate from asymptotic dependence to independence and include the Gaussian dependence as a special case. We show how these new models can be fitted to high threshold exceedances using a censored likelihood approach, and we demonstrate that they provide valuable information about tail characteristics. Our parametric approach outperforms the widely used nonparametric χ and χ statistics often used to guide model choice at an exploratory stage by borrowing strength across locations for better estimation of the asymptotic dependence class. We demonstrate the capacity of our methodology by adequately capturing the extremal properties of wind speed data collected in the Pacific Northwest, US.

In multivariate or spatial extremes, inference for max-stable processes observed at a large collection of locations is a very challenging problem in computational statistics, and current approaches typically rely on less expensive composite likelihoods constructed from small subsets of data. In this work, we explore the limits of modern state-of-the-art computational facilities to perform full likelihood inference and to efficiently evaluate high-order composite likelihoods. With extensive simulations, we assess the loss of information of composite likelihood estimators with respect to a full likelihood approach for some widely-used multivariate or spatial extreme models, we discuss how to choose composite likelihood truncation to improve the efficiency, and we also provide recommendations for practitioners. This article has supplementary material online.

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SUMMARY Genton et al. (2011) investigated the gain in efficiency when triplewise, rather than pairwise, likelihood is used to fit the popular Smith max-stable model for spatial extremes. We generalize their results to the Brown-Resnick model and show that the efficiency gain is substantial only for very smooth processes, which are generally unrealistic in applications.Some key words: Brown-Resnick process; Composite likelihood; Max-stable process; Pairwise likelihood; Smith process; Triplewise likelihood. INTRODUCTIONMax-stable processes are useful for the statistical modelling of spatial extreme events. No finite parametrization of such processes exists, but a spectral representation (de Haan, 1984) aids in constructing models. In a 1990 University of Surrey technical report, R. L. Smith proposed a max-stable model based on deterministic storm profiles which has become popular because it is simple, readily interpreted and easily simulated, but unfortunately it is too inflexible for realism in practice. Another popular model, the Brown-Resnick process, is based on intrinsically stationary log-Gaussian processes, can handle a wide range of dependence structures, and often provides a better fit to data; see, for example, or a 2012 University of North Carolina at Chapel Hill PhD thesis by Soyoung Jeon. Kabluchko et al. (2009) provided further underpinning for this process by showing that that under mild conditions, the BrownResnick process with variogram 2γ(h) = ( h /ρ) α (ρ > 0, 0 < α ≤ 2), where h is the spatial lag, is essentially the only isotropic limit of properly rescaled maxima of Gaussian processes. The Smith model is obtained by taking a Brown-Resnick process with variogram 2γ(h) = h T Σ −1 h for some covariance matrix Σ, corresponding after an affine transformation to taking α = 2, whereas found that 1/2 < α < 1 for the rainfall data they examined.Likelihood inference for max-stable models is difficult, since only the bivariate marginal density functions are known in most cases, and pairwise marginal likelihood is typically used (Padoan et al., 2010;Davison and Gholamrezaee, 2012;Huser and Davison, 2012). This raises the question whether some other approach to inference would be preferable. Genton et al. (2011) derived the general form of the likelihood function for the Smith model and showed that large efficiency gains can arise when fitting it using triplewise, rather than pairwise, likelihood. In this paper we extend their investigation to the Brown-Resnick process and show that for rougher models, more realistic than those considered by Genton et al. (2011), the efficiency gains are much less striking. Thus pairwise likelihood inference provides a good compromise between statistical and computational efficiency in many applications.