Transformed Linear Prediction for Extremes

Abstract

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.