@article{MurphyBarltropetal2023,
   abstract = {In the multivariate setting, estimates of extremal risk measures are important in many contexts, such as environmental planning and structural engineering. In this paper, we propose new estimation methods for extremal bivariate return curves, a risk measure that is the natural bivariate extension to a return level. 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. We apply the methodology to two metocean data sets, with diagnostics indicating generally good performance.},
   author = {C. J. R. Murphy-Barltrop and J. L. Wadsworth and E. F. Eastoe},
   doi = {10.1002/env.2797},
   issn = {1099095X},
   issue = {5},
   journal = {Environmetrics},
   keywords = {dependence modelling,extremes,risk measure},
   month = {8},
   publisher = {John Wiley and Sons Ltd},
   title = {New estimation methods for extremal bivariate return curves},
   volume = {34},
   year = {2023}
  }

@misc{MurphyBarltropetal2024,
      title={Improving estimation for asymptotically independent bivariate extremes via global estimators for the angular dependence function}, 
      author = {C. J. R. Murphy-Barltrop and J. L. Wadsworth and E. F. Eastoe},
      year={2024},
      eprint={2303.13237},
      archivePrefix={arXiv},
      primaryClass={stat.ME}
}

@article{HeffernanTawn2004,
author = {J. E. Heffernan and J. A. Tawn},
title = {A conditional approach for multivariate extreme values (with discussion)},
journal = {Journal of the Royal Statistical Society: Series B (Statistical Methodology)},
volume = {66},
number = {3},
pages = {497-546},
keywords = {Air pollution, Asymptotic independence, Bootstrap, Conditional distribution, Gaussian estimation, Multivariate extreme value theory, Semiparametric modelling},
doi = {https://doi.org/10.1111/j.1467-9868.2004.02050.x},
eprint = {https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9868.2004.02050.x},
abstract = {Multivariate extreme value theory and methods concern the characterization, estimation and extrapolation of the joint tail of the distribution of a d-dimensional random variable. Existing approaches are based on limiting arguments in which all components of the variable become large at the same rate. This limit approach is inappropriate when the extreme values of all the variables are unlikely to occur together or when interest is in regions of the support of the joint distribution where only a subset of components is extreme. In practice this restricts existing methods to applications where d is typically 2 or 3. Under an assumption about the asymptotic form of the joint distribution of a d-dimensional random variable conditional on its having an extreme component, we develop an entirely new semiparametric approach which overcomes these existing restrictions and can be applied to problems of any dimension. We demonstrate the performance of our approach and its advantages over existing methods by using theoretical examples and simulation studies. The approach is used to analyse air pollution data and reveals complex extremal dependence behaviour that is consistent with scientific understanding of the process. We find that the dependence structure exhibits marked seasonality, with extremal dependence between some pollutants being significantly greater than the dependence at non-extreme levels.},
year = {2004}
}

@article{WadsworthTawn2013,
   abstract = {Existing theory for multivariate extreme values focuses upon characterizations of the distributional tails when all components of a random vector, standardized to identical margins, grow at the same rate. In this paper, we consider the effect of allowing the components to grow at different rates, and characterize the link between these marginal growth rates and the multivariate tail probability decay rate. Our approach leads to a whole class of univariate regular variation conditions, in place of the single but multivariate regular variation conditions that underpin the current theories. These conditions are indexed by a homogeneous function and an angular dependence function, which, for asymptotically independent random vectors, mirror the role played by the exponent measure and Pickands' dependence function in classical multivariate extremes. We additionally offer an inferential approach to joint survivor probability estimation. The key feature of our methodology is that extreme set probabilities can be estimated by extrapolating upon rays emanating from the origin when the margins of the variables are exponential. This offers an appreciable improvement over existing techniques where extrapolation in exponential margins is upon lines parallel to the diagonal..},
   author = {J. L. Wadsworth and J. A. Tawn},
   doi = {10.3150/12-BEJ471},
   issn = {13507265},
   issue = {5B},
   journal = {Bernoulli},
   keywords = {Asymptotic independence,Coefficient of tail dependence,Multivariate extreme value theory,Pickands' dependence function,Regular variation},
   month = {11},
   pages = {2689-2714},
   title = {A new representation for multivariate tail probabilities},
   volume = {19},
   year = {2013}
}

@misc{MurphyBarltropWadsworth2024,
      title={Modelling non-stationarity in asymptotically independent extremes}, 
      author={C. J. R. Murphy-Barltrop and J. L. Wadsworth},
      year={2024},
      eprint={2203.05860},
      archivePrefix={arXiv},
      primaryClass={stat.AP}
}

@article{Hill1975,
author = {B. M. Hill},
title = {{A Simple General Approach to Inference About the Tail of a Distribution}},
volume = {3},
journal = {The Annals of Statistics},
number = {5},
publisher = {Institute of Mathematical Statistics},
pages = {1163 -- 1174},
keywords = {Bayesian inference, order statistics, tail of distribution},
year = {1975},
doi = {10.1214/aos/1176343247},
URL = {https://doi.org/10.1214/aos/1176343247}
}

@article{ColesTawn1991,
 ISSN = {00359246},
 doi = {10.1111/j.2517-6161.1991.tb01830.x},
 abstract = {The classical treatment of multivariate extreme values is through componentwise ordering, though in practice most interest is in actual extreme events. Here the point process of observations which are extreme in at least one component is considered. Parametric models for the dependence between components must satisfy certain constraints. Two new techniques for generating such models are presented. Aspects of the statistical estimation of the resulting models are discussed and are illustrated with an application to oceanographic data.},
 author = {S. G. Coles and J. A. Tawn},
 journal = {Journal of the Royal Statistical Society. Series B (Methodological)},
 number = {2},
 pages = {377--392},
 publisher = {[Royal Statistical Society, Wiley]},
 title = {Modelling Extreme Multivariate Events},
 volume = {53},
 year = {1991}
}
