Novel methodology for the estimation of extremal bivariate return curves

Murphy-Barltrop, Callum and Wadsworth, Jennifer and Eastoe, Emma (2023) Novel methodology for the estimation of extremal bivariate return curves. PhD thesis, Lancaster University.

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Abstract

The aim of this thesis is to develop novel methodology for estimating an extreme risk measure, known as a return curve, for bivariate random vectors. In doing this, we also aim to develop novel techniques for estimating the extremal dependence structure of bivariate random vectors, and then to compare these techniques to existing methodology. In many practical applications, understanding joint extreme risks from pairs of random variables is crucial for ensuring robust risk analyses and informed decision making. Return curves provide a means of both quantifying and visualising such risks. However, estimation of these curves has not been well studied, particularly in the case when data exhibits asymptotic independence. Furthermore, under the influence of climate change, the joint extremal behaviour for pairs of environmental variables is likely to change; techniques are required to ensure such trends are captured when estimating return curves. We first propose a range of novel estimation methods for return curves; unlike several existing techniques, our estimates are based on bivariate extreme value models that can capture both key forms of extremal dependence. We devise tools for validating return curve estimates, as well as representing their uncertainty, and compare a selection of curve estimation techniques through simulation studies. Curve estimates are obtained for two metocean data sets, with diagnostics indicating generally good performance. In the context of extremes, few methods have been proposed for modelling trends in extremal dependence, even though capturing this feature is important for quantifying joint tail behaviour. Motivated by observed dependence trends in data from the UK Climate Projections, we propose a novel semi-parametric modelling framework for non-stationary, bivariate extremal dependence structures. This framework allows us to capture a wide variety of dependence trends for datasets exhibiting asymptotic independence. We compare our model to an existing technique through a simulation study, obtaining competitive results over a range of extremal dependence structures. When applied to a climate projection dataset, our model is able to capture observed dependence trends and, in combination with models for marginal non-stationarity, can be used to produce estimates of return curves in future climates. Whilst asymptotic independence is frequently observed in practice, the majority of approaches for bivariate extremes are based on the framework of regular variation. In practice, this is problematic since this framework is is unable to accurately extrapolate into the joint tail for data sets exhibiting this class of extremal dependence. Motivated by this shortcoming, we introduce a range of novel estimators for the so-called `angular dependence function', a quantity which summarises the dependence structure for asymptotically independent variables. We compare the proposed estimators to existing techniques through a systematic simulation study, obtaining competitive results in many cases. The proposed methodology is also applied to river flow data from the north of England, UK, and used to obtain return curve estimates for different pairs of gauge sites.