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.

Summary1. Species distribution models are increasingly used to address conservation questions, so their predictive capacity requires careful evaluation. Previous studies have shown how individual factors used in model construction can affect prediction. Although some factors probably have negligible effects compared to others, their relative effects are largely unknown. 2. We introduce a general 'virtual ecologist' framework to study the relative importance of factors involved in the construction of species distribution models. 3. We illustrate the framework by examining the relative importance of five key factors -a missing covariate, spatial autocorrelation due to a dispersal process in presences/absences, sample size, sampling design and modelling technique -in a real study framework based on virtual plants in a mountain landscape at regional scale, and show that, for the parameter values considered here, most of the variation in prediction accuracy is due to sample size and modelling technique. Contrary to repeatedly reported concerns, spatial autocorrelation has only comparatively small effects. 4. This study shows the importance of using a nested statistical framework to evaluate the relative effects of factors that may affect species distribution models.

Recent advances in extreme value theory have established ℓ-Pareto processes as the natural limits for extreme events defined in terms of exceedances of a risk functional. Here we provide methods for the practical modelling of data based on a tractable yet flexible dependence model. We introduce the class of elliptical ℓ-Pareto processes, which arise as the limit of threshold exceedances of certain elliptical processes characterized by a correlation function and a shape parameter. An efficient inference method based on maximizing a full likelihood with partial censoring is developed. Novel procedures for exact conditional and unconditional simulation are proposed. These ideas are illustrated using precipitation extremes in Switzerland.

Employing the framework of regular variation, we propose two decompositions which help to summarize and describe high-dimensional tail dependence. Via transformation, we define a vector space on the positive orthant, yielding the notion of basis. With a suitably-chosen transformation, we show that transformed-linear operations applied to regularly varying random vectors preserve regular variation. Rather than model regular variation's angular measure, we summarize tail dependence via a matrix of pairwise tail dependence metrics. This matrix is positive semidefinite, and eigendecomposition allows one to interpret tail dependence via the resulting eigenbasis. Additionally this matrix is completely positive, and a resulting decomposition allows one to easily construct regularly varying random vectors which share the same pairwise tail dependencies. We illustrate our methods with Swiss rainfall data and financial return data.

[1] We propose an approach to spatial modeling of extreme rainfall, based on max-stable processes fitted using partial duration series and a censored threshold likelihood function. The resulting models are coherent with classical extreme-value theory and allow the consistent treatment of spatial dependence of rainfall using ideas related to those of classical geostatistics. We illustrate the ideas through data from the Val Ferret watershed in the Swiss Alps, based on daily cumulative rainfall totals recorded at 24 stations for four summers, augmented by a longer series from nearby. We compare the fits of different statistical models appropriate for spatial extremes, select that best fitting our data, and compare return level estimates for the total daily rainfall over the stations. The method can be used in other situations to produce simulations needed for hydrological models, and in particular, for the generation of spatially heterogeneous extreme rainfall fields over catchments.

Spatial modeling of rare events has obvious applications in the environmental sciences and is crucial when assessing the effects of catastrophic events (such as heatwaves or widespread flooding) on food security and on the sustainability of societal infrastructure. Although classical geostatistics is largely based on Gaussian processes and distributions, these are not appropriate for extremes, for which maxstable and related processes provide more suitable models. This paper provides a brief overview of current work on the statistics of spatial extremes, with an emphasis on the consequences of the assumption of max-stability. Applications to winter minimum temperatures and daily rainfall are described.

We present a method for drawing isolines indicating regions of equal joint exceedance probability for bivariate data. The method relies on bivariate regular variation, a dependence framework widely used for extremes. This framework enables drawing isolines corresponding to very low exceedance probabilities and these lines may lie beyond the range of the data. The method we utilize for characterizing dependence in the tail is largely nonparametric. Furthermore, we extend this method to the case of asymptotic independence and propose a procedure which smooths the transition from asymptotic independence in the interior to the first-order behavior on the axes. We propose a diagnostic plot for assessing isoline estimate and choice of smoothing, and a bootstrap procedure to visually assess uncertainty.

The Brown-Resnick max-stable process has proven to be wellsuited for modeling extremes of complex environmental processes, but in many applications its likelihood function is intractable and inference must be based on a composite likelihood, thereby preventing the use of classical Bayesian techniques. In this paper we exploit a case in which the full likelihood of a Brown-Resnick process can be calculated, using componentwise maxima and their partitions in terms of individual events, and we propose two new approaches to inference. The first estimates the partitions using declustering, while the second uses random partitions in a Markov chain Monte Carlo algorithm. We use these approaches to construct a Bayesian hierarchical model for extreme low temperatures in northern Fennoscandia.