We consider the problem of performing prediction when observed values are at their highest levels. We construct an inner product space of nonnegative random variables from transformed-linear combinations of independent regularly varying random variables. The matrix of inner products corresponds to the tail pairwise dependence matrix, which summarizes tail dependence. The projection theorem yields the optimal transformed-linear predictor, which has the same form as the best linear unbiased predictor in non-extreme prediction. We also construct prediction intervals based on the geometry of regular variation. We show that these intervals have good coverage in a simulation study as well as in two applications; prediction of high pollution levels, and prediction of large financial losses.

翻译：我们考虑了在观测值处于最高水平时进行预测的问题。我们从独立、经常变化的随机变量的变式线性组合中构建了一个非负随机变量的内部产品空间。内部产品矩阵与尾端双向依赖性矩阵相对应,该矩阵总结了尾尾端依赖性。预测的定理产生最佳的流线性预测器,其形式与非极端预测中的最佳线性无偏向预测器相同。我们还根据定期变异的几何方法构建了预测间隔。我们显示这些间隔在模拟研究中和两个应用中都有很好的覆盖;高污染水平预测,以及重大财政损失预测。