This thesis develops novel statistical methodologies to bring closer the fields of extreme value theory and causality. It revolves around two independent axes of research.
The first axis studies causal discovery for extreme events, where one can infer the causal structure of a system by exploiting the signal in the tails of the variables.
In the first project, we introduce a causal coefficient that identifies the causal relationship of heavy-tailed pairs of variables. Then, we propose a computationally highly efficient algorithm based on this causal tail coefficient to recover the causal order of a set of variables. Finally, we compare our method to other well-established and non-extremal approaches in causal discovery on synthetic and real data.
The second axis of research develops flexible predictive models for extremes and distribution generalization.
The second project of this thesis develops a quantile regression method to estimate extreme quantiles given a large set of predictors. Our method combines the flexibility of the random forests with the extrapolation guarantees of the generalized Pareto distribution. In simulations, our method is competitive with both classical quantile regression methods and existing regression approaches from extreme value theory. Finally, we apply our methodology to extreme quantile prediction for U.S. wage data.
The third project of this thesis studies the problem of distribution generalization from a causal perspective. We assume the data comes from different environments that shift the mean of the predictors so that the training and test distributions are different. We model distributional shifts with the concept of causal intervention. Here, we propose a method to learn a nonparametric function with invariant predictions across environments and as predictive as possible, defined as the invariant most predictive (IMP) function. We show identification of the IMP, provide minimax guarantees over unseen environments over the class of square-integrable functions, and propose an adaptation of the regression tree algorithm to learn the IMP function nonparametrically in large dimensions.
