Robust Elicitable Functionals

Elicitable functionals and (strictly) consistent scoring functions are of interest due to their utility of determining (uniquely) optimal forecasts, and thus the ability to effectively backtest predictions. However, in practice, assuming that a distribution is correctly specified is too strong a belief to reliably hold. To remediate this, we incorporate a notion of statistical robustness into the framework of elicitable functionals, meaning that our robust functional accounts for “small” misspecifications of a baseline distribution. Specifically, we propose a robustified version of elicitable functionals by using the Kullback-Leibler divergence to quantify potential misspecifications from a baseline distribution. We show that the robust elicitable functionals admit unique solutions lying at the boundary of the uncertainty region, and provide conditions for existence and uniqueness. Since every elicitable functional possesses infinitely many scoring functions, we propose the class of b-homogeneous strictly consistent scoring functions, for which the robust functionals maintain desirable statistical properties. We show the applicability of the robust elicitable functional in several examples: in a reinsurance setting and in robust regression problems.

💡 Research Summary

The paper introduces a novel framework called the Robust Elicitable Functional (REF) that merges the concept of elicitability with distributional robustness measured by the Kullback‑Leibler (KL) divergence. Traditional risk measures either assume a perfectly known baseline distribution or adopt a worst‑case approach that maximizes the risk functional over an uncertainty set. Both approaches are problematic in practice because data are often outdated, sparse, or contaminated, making the baseline distribution uncertain.

The authors define an uncertainty set Qε = { Q ≪ P : DKL(Q‖P) ≤ ε } around a baseline probability measure P, where ε ≥ 0 quantifies the tolerance for misspecification. For any elicitable functional R with a strictly consistent scoring function S, the REF is defined as

R_S(Y) = arg min_{z∈A} sup_{Q∈Qε} E_Q