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Publication# Functional peaks-over-threshold analysis

Résumé

Peaks-over-threshold analysis using the generalised Pareto distribution is widely applied in modelling tails of univariate random variables, but much information may be lost when complex extreme events are studied using univariate results. In this paper, we extend peaks-over-threshold analysis to extremes of functional data. Threshold exceedances defined using a functional r are modelled by the generalised r-Pareto process, a functional generalisation of the generalised Pareto distribution that covers the three classical regimes for the decay of tail probabilities, and that is the only possible continuous limit for r-exceedances of a properly rescaled process. We give construction rules, simulation algorithms and inference procedures for generalised r-Pareto processes, discuss model validation and apply the new methodology to extreme European windstorms and heavy spatial rainfall.

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xtreme value analysis is concerned with the modelling of extreme events such as floods and heatwaves, which can have large impacts. Statistical modelling can be useful to better assess risks even if, due to scarcity of measurements, there is inherently very large residual uncertainty in any analysis. Driven by the increase in environmental databases, spatial modelling of extremes has expanded rapidly in the last decade. This thesis presents contributions to such analysis.
The first chapter is about likelihood-based inference in the univariate setting and investigates the use of bias-correction and higher-order asymptotic methods for extremes, highlighting through examples and illustrations the unique challenge posed by data scarcity. We focus on parametric modelling of extreme values, which relies on limiting distributional results and for which, as a result, uncertainty quantification is complicated. We find that, in certain cases, small-sample asymptotic methods can give improved inference by reducing the error rate of confidence intervals. Two data illustrations, linked to assessment of the frequency of extreme rainfall episodes in Venezuela and the analysis of survival of supercentenarians, illustrate the methods developed.
In the second chapter, we review the major methods for the analysis of spatial extremes models. We highlight the similarities and provide a thorough literature review along with novel simulation algorithms. The methods described therein are made available through a statistical software package.
The last chapter focuses on estimation for a Bayesian hierarchical model derived from a multivariate generalized Pareto process. We review approaches for the estimation of censored components in models derived from (log)-elliptical distributions, paying particular attention to the estimation of a high-dimensional Gaussian distribution function via Monte Carlo methods. The impacts of model misspecification and of censoring are explored through extensive simulations and we conclude with a case study of rainfall extremes in Eastern Switzerland.

Emeric Rolland Georges Thibaud

This thesis proposes some contributions to the spatial modelling of species distributions and extreme values. Predictive models are increasingly used to model the distribution of species and to estimate the potential effects of global change on biodiversity. Although species distribution models are now widely used, the effects of several factors on their prediction accuracy are mostly unknown. We introduce a method to measure the relative impact of factors on the accuracy of species distribution models. Our approach allows us to identify factors that require more control in the construction of predictive models. The ideas are illustrated by application to plant species in the Swiss Alps. The modelling of spatial extremes, or rare events, is central for the assessment of risks associated to disastrous environmental events. Methods for modelling extremes of time series are well-established, but efficient methods for spatial modelling are still in full development. We investigate the use of asymptotic dependence and independence models based on max-stable processes and their differences in practice. We show how these different models can be used to model threshold exceedances, and we introduce a more efficient approach based on Pareto processes. Finally, we construct a flexible Bayesian hierarchical model for spatial extremes. Our methods are illustrated by applications to rainfall in Switzerland and low temperatures in Finland.

Jonathan A. Tawn, Jennifer Lynne Wadsworth

Max-stable processes arise as the only possible nontrivial limits for maxima of affinely normalized identically distributed stochastic processes, and thus form an important class of models for the extreme values of spatial processes. Until recently, inference for max-stable processes has been restricted to the use of pairwise composite likelihoods, due to intractability of higher-dimensional distributions. In this work we consider random fields that are in the domain of attraction of a widely used class of max-stable processes, namely those constructed via manipulation of log-Gaussian random functions. For this class, we exploit limiting d-dimensional multivariate Poisson process intensities of the underlying process for inference on all d-vectors exceeding a high marginal threshold in at least one component, employing a censoring scheme to incorporate information below the marginal threshold. We also consider the d-dimensional distributions for the equivalent max-stable process, and perform full likelihood inference by exploiting the methods of Stephenson & Tawn (2005), where information on the occurrence times of extreme events is shown to dramatically simplify the likelihood. The Stephenson-Tawn likelihood is in fact simply a special case of the censored Poisson process likelihood. We assess the improvements in inference from both methods over pairwise likelihood methodology by simulation.