arXiv subjects

Benjamin Bobbia

Publications and source records attributed to Benjamin Bobbia.

8 recordsLinked to original sources

arXiv2026

On the optimal prediction of extreme events

The prediction of the extremely large values of a response variable $Y$ in terms of a vector of covariates $X=(X_i)_{i=1}^d$ is a fundamental problem arising in many scientific and engineering domains. The scarcity of data in the extremes makes the optimal solution of this problem of particular importance. The optimal predictors of such events can be explicitly characterized in just a few cases and it is of fundamental practical and theoretical interest to develop optimal estimators over large classes of models and predictors. In this work, the focus is on the case where $(Y,X)$ have a multivariate regularly varying distribution and one seeks an optimal predictor expressed as a positive homogeneous function $h(X)$ of the covariates. The asymptotic prediction precision in this setting coincides with the tail-dependence coefficient $\lambda(Y,h(X))$ and it can be expressed as an integral functional of the associated angular measure of $(Y,X)$. Thus, finding asymptotically optimal homogeneous predictors amounts to solving a variational problem. We obtain a general solution to this problem, which is expressed in terms of a non-extreme conditional quantile of a tilted distribution derived from the angular measure. This leads to a general inference methodology for the optimal predictors in the peaks-over-threshold framework form extreme value theory. We establish the universal consistency for these estimators over large classes of angular measures. A general-purpose implementation of the resulting inference procedure is shown to work remarkably well against optimal oracle estimators, as well as in the challenging problem of extreme solar flare prediction.

math.ST↗

arXiv2026

Tessellation Localized Transfer learning for nonparametric regression

Transfer learning aims to improve performance on a target task by leveraging information from related source tasks. We propose a nonparametric regression transfer learning framework that explicitly models heterogeneity in the source-target relationship. Our approach relies on a local transfer assumption: the covariate space is partitioned into finitely many cells such that, within each cell, the target regression function can be expressed as a low-complexity transformation of the source regression function. This localized structure enables effective transfer where similarity is present while limiting negative transfer elsewhere. We introduce estimators that jointly learn the local transfer functions and the target regression, together with fully data-driven procedures that adapt to unknown partition structure and transfer strength. We establish sharp minimax rates for target regression estimation, showing that local transfer can mitigate the curse of dimensionality by exploiting reduced functional complexity. Our theoretical guarantees take the form of oracle inequalities that decompose excess risk into estimation and approximation terms, ensuring robustness to model misspecification. Numerical experiments illustrate the benefits of the proposed approach.

math.ST↗

arXiv2025

Variance-reduced extreme value index estimators using control variates in a semi-supervised setting

The estimation of the Extreme Value Index (EVI) is fundamental in extreme value analysis but suffers from high variance due to reliance on only a few extreme observations. We propose a control variates based transfer learning approach in a semi-supervised framework, where a small set of coupled target and source observations is combined with abundant unpaired source data. By expressing the Hill estimator of the target EVI as a ratio of means, we apply approximate control variates to both numerator and denominator, with jointly optimized coefficients that guarantee variance reduction without introducing bias. We show theoretically and through simulations that the asymptotic relative variance reduction of the transferred Hill estimator is proportional to the tail dependence between the target and source variables and independent of their EVI values. Thus, substantial variance reduction can be achieved even without similarity in tail heaviness of the target and source distributions. The proposed approach can be extended to other EVI estimators expressed with ratio of means, as demonstrated on the moment estimator. The practical value of the proposed method is illustrated on multi-fidelity water surge and ice accretion datasets.

stat.ME↗

arXiv2025

Control variates for variance-reduced ratio of means estimators

The control variates method is a classical variance reduction technique for Monte Carlo estimators that exploits correlated auxiliary variables without introducing bias. In many applications, the quantity of interest can be expressed as a ratio of expectations. We propose a variance-reduced estimator for such ratios, which applies control variates to both the numerator and the denominator. The control variates coefficients are optimized jointly to minimize the approximated variance of the resulting estimator. This approach guarantees variance reduction and naturally extends to approximate control variates. Simulation studies show significant variance reduction, particularly when correlations between variables and control variates are strong. The practical value of the method is illustrated on multi-fidelity applications: estimating a proportion in an aircraft design use case and a conditional value-at-risk in an electromagnetic dataset.

math.ST↗

arXiv2025

Do you understand epistemic uncertainty? Think again! Rigorous frequentist epistemic uncertainty estimation in regression

Quantifying model uncertainty is critical for understanding prediction reliability, yet distinguishing between aleatoric and epistemic uncertainty remains challenging. We extend recent work from classification to regression to provide a novel frequentist approach to epistemic and aleatoric uncertainty estimation. We train models to generate conditional predictions by feeding their initial output back as an additional input. This method allows for a rigorous measurement of model uncertainty by observing how prediction responses change when conditioned on the model's previous answer. We provide a complete theoretical framework to analyze epistemic uncertainty in regression in a frequentist way, and explain how it can be exploited in practice to gauge a model's uncertainty, with minimal changes to the original architecture.

stat.ML↗

arXiv2024

Active Learning for Regression based on Wasserstein distance and GroupSort Neural Networks

This paper addresses a new active learning strategy for regression problems. The presented Wasserstein active regression model is based on the principles of distribution-matching to measure the representativeness of the labeled dataset. The Wasserstein distance is computed using GroupSort Neural Networks. The use of such networks provides theoretical foundations giving a way to quantify errors with explicit bounds for their size and depth. This solution is combined with another uncertainty-based approach that is more outlier-tolerant to complete the query strategy. Finally, this method is compared with other classical and recent solutions. The study empirically shows the pertinence of such a representativity-uncertainty approach, which provides good estimation all along the query procedure. Moreover, the Wasserstein active regression often achieves more precise estimations and tends to improve accuracy faster than other models.

cs.LG↗

arXiv2020

Extreme quantile regression in a proportional tail framework

We revisit the model of heteroscedastic extremes initially introduced by Einmahl et al. (JRSSB, 2016) to describe the evolution of a non stationary sequence whose extremes evolve over time and adapt it into a general extreme quantile regression framework. We provide estimates for the extreme value index and the integrated skedasis function and prove their asymptotic normality. Our results are quite similar to those developed for heteroscedastic extremes but with a different proof approach emphasizing coupling arguments. We also propose a pointwise estimator of the skedasis function and a Weissman estimator of the conditional extreme quantile and prove the asymptotic normality of both estimators.

math.ST↗

arXiv2019

The coupling method in extreme value theory

A coupling method is developed for univariate extreme value theory , providing an alternative to the use of the tail empirical/quantile processes. Emphasizing the Peak-over-Threshold approach that approximates the distribution above high threshold by the Generalized Pareto distribution, we compare the empirical distribution of exceedances and the empirical distribution associated to the limit Generalized Pareto model and provide sharp bounds for their Wasser-stein distance in the second order Wasserstein space. As an application , we recover standard results on the asymptotic behavior of the Hill estimator, the Weissman extreme quantile estimator or the probability weighted moment estimators, shedding some new light on the theory.

math.ST↗