Search

Extreme events are responsible for huge material damage and are costly in terms of their human and economic impacts. They strike all facets of modern society, such as physical infrastructure and insurance companies through environmental hazards, banking and finance through stock market crises, and the internet and communication systems through network and server overloads. It is thus of increasing importance to accurately assess the risk of extreme events in order to mitigate them. Extreme value theory is a statistical approach to extrapolation of probabilities beyond the range of the data, which provides a robust framework to learn from an often small number of recorded extreme events. In this thesis, we consider a conditional approach to modelling extreme values that is more flexible than standard models for simultaneously extreme events. We explore the subasymptotic properties of this conditional approach and prove that in specific situations its finite-sample behaviour can differ significantly from its limit characterisation. For modelling extremes in time series with short-range dependence, the standard peaks-over-threshold method relies on a pre-processing step that retains only a subset of observations exceeding a high threshold and can result in badly-biased estimates. This method focuses on the marginal distribution of the extremes and does not estimate temporal extremal dependence. We propose a new methodology to model time series extremes using Bayesian semiparametrics and allowing estimation of functionals of clusters of extremes. We apply our methodology to model river flow data in England and improve flood risk assessment by explicitly describing extremal dependence in time, using information from all exceedances of a high threshold. We develop two new bivariate models which are based on the conditional tail approach, and use all observations having at least one extreme component in our inference procedure, thus extracting more information from the data than existing approaches. We compare the efficiency of these models in a simulation study and discuss generalisations to higher-dimensional setups. Existing models for extremes of Markov chains generally rely on a strong assumption of asymptotic dependence at all lags and separately consider marginal and joint features. We introduce a more flexible model and show how Bayesian semiparametrics can provide a suitable framework allowing simultaneous inference for the margins and the extremal dependence structure, yielding efficient risk estimates and a reliable assessment of uncertainty.

Linear regression summarises the link between a variable of interest and one or several explanatory variables through a parametric relationship. Hastie and Tibshirani (1986) extended this approach by allowing a non-parametric description of the dependence between the variable of interest and the explanatory variables. In this report, we review the notion of smooths, present different approaches to selection of the smoothing parameter, and describe several spline bases, in the lines of Wood (2006). We show evidence of smoking as a cause of deaths from larynx cancer by applying Gaussian Markov random fields and thin plate regression splines on data covering 544 German districts from 1986 to 1990.

Heavy-tail phenomena are common in real-life data; the finance and insurance industries, telecommunications, and environment-related events offer typical examples of such phenomena. We focus on the particular topic of the extremogram, for which Davis and Mikosh (2009) present an empirical estimator. We propose the use of a semi-parametric model which allows extrapolation beyond the range of the data and flexible enough to cover any type of extremal dependence.

Extreme events can be statistically characterised as excesses of a high threshold. Inference in this case has to account for dependence between excesses. The peaks over threshold approach suggests pre-processing the series by defining clusters of successive observations and making inference only on maxima of those clusters, which can be assumed independent. There exists however no good method for specifying the clusters, causing quantiles for long return periods to be potentially badly biased. The peaks over threshold method uses asymptotic results as approximation at finite levels; our goal is to focus on the subasymptotic model suggested by Eastoe and Tawn (2012) to develop new Bayesian semiparametric techniques to properly approach the problem, thus getting an estimation of the full excess distribution uncertainty. A simulation study shows the instability in estimated quantiles based on the peaks over threshold method compared to the subasymptotic model across different cluster definitions. An application on river peakflows is also presented and shows how the subasymptotic model, fitted with the novel semiparametric Bayesian method, can be applied to real data. This work is divided into three main parts: an introductory part which provides an insight into multivariate extremes, with a particular attention to special kinds of dependence involved in this framework. In the second part we introduce a conditional multivariate model and develop a semiparametric Gibbs sampler to fit this particular model. The last part deals with the subasymptotic approach, meant to model short-range dependence of excesses over a threshold. We discuss further improvements and alternatives in the last section.

Frequentist and Bayesian approaches to statistics have long been seen as incompatible, but recent work has been done to try and unify them (Bayarri and Berger, 2004; Efron, 2005). Empirical Bayes, approximate Bayesian analysis, and the matching prior approach are examples of methods where prior elicitation is driven by a frequentist interpretation of the data. In this report, we review some standard frequentist and Bayesian model selection techniques and describe how objective Bayes theory can help in this decision-oriented framework, typically by enabling consideration of uncertainty in the model-building process. Objective Bayes mehtods are then used and compared with standard model selection methods on data from the financial sector in Switzerland, Greece, and the United States during 1999 to 2013; results show broad agreement between the methods, but conclusions are less clear-cut in the objective Bayes framework.