Empirical Likelihood for Nonsmooth Functionals

2026-03-29

AI Summary

This work addresses the limitations of conventional empirical likelihood methods, which rely on smoothness assumptions and fail to provide reliable inference for nonsmooth functionals—such as optimal value functionals in policy evaluation—particularly when the optimum is non-unique. To overcome this, the authors propose a geometric bootstrap empirical likelihood approach that reformulates the profile likelihood as the distance from the mean of estimating scores to a specific level set. By circumventing Taylor expansions and directly leveraging the convex optimization structure inherent in nonsmooth functionals, this method integrates empirical likelihood, geometric analysis, and an adaptive multiplier bootstrap. The resulting framework enables valid statistical inference for nonsmooth functionals, yielding accurately calibrated confidence intervals even in complex scenarios and substantially improving inferential reliability.

Abstract

Empirical likelihood is an attractive inferential framework that respects natural parameter boundaries, but existing approaches typically require smoothness of the functional and miscalibrate substantially when these assumptions are violated. For the optimal-value functional central to policy evaluation, smoothness holds only when the optimum is unique -- a condition that fails exactly when rigorous inference is most needed where more complex policies have modest gains. In this work, we develop a bootstrap empirical likelihood method for partially nonsmooth functionals. Our analytic workhorse is a geometric reduction of the profile likelihood to the distance between the score mean and a level set whose shape (a tangent cone given by nonsmoothness patterns) determines the asymptotic distribution. Unlike the classical proof technology based on Taylor expansions on the dual optima, our geometric approach leverages properties of a deterministic convex program and can directly apply to nonsmooth functionals. Since the ordinary bootstrap is not valid in the presence of nonsmoothness, we derive a corrected multiplier bootstrap approach that adapts to the unknown level-set geometry.

Problem

• empirical likelihood
• nonsmooth functionals
• policy evaluation
• optimal-value functional
• statistical inference

Innovation

• empirical likelihood
• nonsmooth functionals
• bootstrap
• tangent cone
• policy evaluation