Modern risk assessment in many areas of interest require estimation of the extremal behaviour of sums of random variables. We derive the first order upper-tail behaviour of the weighted sum of bivariate random variables under weak assumptions on their marginal distributions and their copula. The extremal behaviour of the marginal variables is characterised by the generalised Pareto distribution and their extremal dependence is characterised using subclasses of the limiting representations of Ledford and Tawn (1997) and Heffernan and Tawn (2004), which describe both components being jointly extreme, and the behaviour of one component conditional on the other component being large, respectively. We find that the upper are both positive or have different signs, the upper tail behaviour of the aggregate is driven by different factors dependent on the signs of the marginal shape parameters; if they are both negative, the extremal behaviour of the aggregate is driven by both marginal shape parameters and the coefficient of asymptotic independence Ledford and Tawn (1996); if they are both positive or have different signs, the upper-tail behaviour of the aggregate is driven solely be the largest marginal shape. We also derive the aggregate upper-tail behaviour for some well known copulae which reveals further insight into the tail structure when the copula falls outside the conditions for the subclasses of the limiting dependence representations. We conduct inference on the upper-tail of aggregates of gridded UK precipitation and temperature data to illustrate that our limit results provide good approximations in practice.
翻译:在许多感兴趣的领域,现代风险评估要求估计随机变量总量的极端行为。我们根据对边际分布和千叶草质的脆弱假设,得出双变随机变量加权总和的第一顺序高尾行为。边际变量的极端行为特征是泛泛Pareto分布及其极端依赖性,其特征是使用Ledford和Tawn(1997年)、Heffernan和Tawn (2004年)这两个部分的限制性表现的子类,这两个部分都描述了共同的极端,一个部分的行为分别以另一部分的大为条件。我们发现,上层是正的,或者有不同的迹象,总体的上尾部行为是由取决于边际形状参数迹象的不同因素驱动的。如果它们都是负面的,则总体的极端行为是由边际形状参数和偏差独立Ledford和Tawn(1996年)的系数所驱动的;如果这两个部分是正面的或有不同迹象,则一个部分的上层是最大的边形。我们发现,总量的上尾部行为只是最边缘的形状。我们还从某些已知的尾尾部行为中得出了某些已知的底底部行为,在COL的底部表现。当我们所了解的COL的底部的底部显示的底部的底部的底部,则会的底部显示为英国的底部。