We address the problem of prediction for extreme observations by proposing an extremal linear prediction method. We construct an inner product space of nonnegative random variables derived from transformed-linear combinations of independent regularly varying random variables. Under a reasonable modeling assumption, the matrix of inner products corresponds to the tail pairwise dependence matrix, which can be easily estimated. We derive the optimal transformed-linear predictor via the projection theorem, which yields a predictor with the same form as the best linear unbiased predictor in non-extreme settings. We quantify uncertainty for prediction errors by constructing prediction intervals based on the geometry of regular variation. We demonstrate the effectiveness of our method through a simulation study and its applications to predicting high pollution levels, and extreme precipitation.

翻译：本文通过提出一种极值线性预测方法，解决极端观测值的预测问题。我们构建了一个由独立正则变化随机变量的变换线性组合导出的非负随机变量内积空间。在合理的建模假设下，内积矩阵对应于尾部成对依赖矩阵，该矩阵易于估计。我们通过投影定理推导出最优变换线性预测器，其形式与非极端场景中的最佳线性无偏预测器相同。基于正则变化的几何特性，我们构建预测区间以量化预测误差的不确定性。通过模拟研究及其在高污染水平预测与极端降水预测中的应用，我们验证了该方法的有效性。