Abstract
Probabilistic and statistical aspects of extremes of univariate processes have been extensively studied, and recent developments in extremes have focused on multivariate theory and its application. Multivariate extreme value theory encompasses two separate aspects: marginal features, which may be handled by standard univariate methods, and dependence features. Both will be examined in this study. First we focus on testing independence in multivariate extremes. All existing score tests of independence in multivariate extreme values have non-regular properties that arise due to violations of the usual regularity conditions of maximum likelihood. Some of these violations may be dealt with using standard techniques, for example when independence corresponds to a boundary point of the parameter space of the underlying model. However, another type of regularity violation, the infinite second moment of the score function, is more difficult to deal with and has important consequences for applications, resulting in score statistics with non-standard normalisation and poor rates of convergence. We propose a likelihood based approach that provides asymptotically normal score tests of independence with regular normalisation and rapid convergence. The resulting tests are straightforward to implement and are beneficial in practical situations with realistic amounts of data. A fundamental issue in applied multivariate extreme value (MEV) analysis is modelling dependence within joint tail regions. The primary aim of the remainder of this thesis is to develop a pseudo-polar framework for modelling extremal dependence that extends the existing classical results for multivariate extremes to encompass asymptotically independent tails. Accordingly, a constructional procedure for obtaining parametric asymptotically independent joint tail models is developed. The practical application of this framework is analysed through applications to bivariate simulated and environmental data, and joint estimation of dependence and marginal parameters via likelihood methodology is detailed. Inference under our models is examined and tests of extremal asymptotic independence and asymmetry are derived which are useful for model selection. In contrast to the classical MEV approach, which concentrates on the distribution of the normalised componentwise maxima, our framework is based on modelling joint tails and focuses directly on the tail structure of the joint survivor function. Consequently, this framework provides significant extensions of both the theoretical and applicable tools of joint tail modelling. Analogous point process theory is developed and the classical componentwise maxima result for multivariate extremes is extended to the asymptotically independent case. Finally, methods for simulating from two of our bivariate parametric models are provided.