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- ItemDistribution theory and inference for bivariate extremes(Stellenbosch : Stellenbosch University, 2024-03) van Tonder, Jana; Steyn, Matthys Lucas; de Wet, Tertius; Stellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
Show more ENGLISH SUMMARY: Various scenarios exist where the interest is in the modelling and prediction of rare or extreme events. Extreme value theory is an important branch of statistics, where limit theory is used to analyse extremes and to estimate the tail of the underlying distribution. Extreme value theory is the most developed for the univariate case, i.e. modelling the extremes of only a single variable. In many scenarios, however, more than one variable has an effect on the probability of occurrence of extreme events. In such cases, multivariate extreme value theory will play a valuable role in the modelling procedure by taking into account the joint effect of multivariate extremes. In this thesis, the focus will be on bivariate extreme value theory, i.e., multivariate extreme value theory restricted to two dimensions. Two approaches will be considered: (1) componentwise maxima and (2) a pair of random variables above a large threshold vector. A mathematical derivation of the limiting distribution of normalised componentwise maxima, called the bivariate extreme value distribution, will be given. For the threshold exceedance approach, it will be shown how the underlying distribution can be approximated by the bivariate extreme value distribution at transformed points. Unfortunately, no parametric form exists for the bivariate extreme value distribution. However, the distribution can be expressed in terms of the two marginal distributions and a dependence function. The latter is important in characterising the dependence structure of the distribution. Various characterisations are proposed in the literature. A few popular dependence functions will be discussed. It will also be shown how they are related through appropriate transformations. Since dependence plays an important role in bivariate extreme value theory, different measures of extremal dependence will be examined. For an independent and identically distributed random bivariate sample with asymptotic dependence between the two variables, it will be shown how the limit theory, based on the bivariate extreme value distribution, can be applied and how inference can be performed. Different ways of estimating the dependence structure of the bivariate extreme value distribution will be described, which include parametric and non-parametric techniques. When data exhibit asymptotic independence, the bivariate extreme value distribution is not suitable to use in the modelling procedure. Therefore, other models will be explored which better describe the tail of an asymptotically independent distribution. For illustration, the above-mentioned methods will be applied to two South African bivariate environmental datasets. For further interpretation and visualisation, graphs of the estimated distributions and quantile curves will also be given. Finally, it will be demonstrated that an asymptotic dependent model can lead to an overestimation of the joint exceedance probability when working in the tail of an asymptotic independent distribution, which agrees with the findings in the literature.Show more