arXiv:2410.05517v4 Announce Type: replace Abstract: An extreme dimension reduction method is proposed, extending the Extreme-PLS approach to the discretized functional framework, where the covariate lies in the infinite-dimensional Hilbert space ($L^2([0,1])$) but is partially observed on a dense time grid. The ideas are partly borrowed from both Partial Least-Squares (PLS) and Sliced Inverse Regression (SIR) techniques. Accordingly, the method relies on the projection of the covariate onto a subspace and maximizes the covariance between its projection and the response conditionally on an extreme event capturing tail information. The covariate and the heavy-tailed response are supposed to be linked through a nonlinear inverse single-index model, and the objective is to infer the index in this regression framework. A new family of estimators is proposed, and its asymptotic consistency with convergence rates under the model is established. Assuming mild conditions on the noise, most of the assumptions are stated in terms of regular variation, unlike those commonly imposed for SIR and single-index regression. In addition, the theoretical analysis is expanded to include a model-free consistency result in probability for the empirical tail-moments in a general separable Hilbert space. Finally, the results are illustrated through a finite-sample study with synthetic functional data and high-frequency financial data, highlighting the effectiveness of the dimension reduction in capturing tail dependence and supporting extreme risk management.