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：我们考虑在观测值处于最高水平时进行预测的问题。基于独立正则变化随机变量的变换线性组合，我们构造了一个非负随机变量的内积空间。该内积矩阵对应于尾部成对依赖矩阵，它总结了尾部依赖关系。投影定理给出了最优变换线性预测器，其形式与非极值预测中的最佳线性无偏预测器相同。我们还基于正则变化几何性质构造了预测区间。模拟研究及两项应用（高污染水平预测与重大金融损失预测）表明，这些区间具有较好的覆盖率。