The Role of Empirical Likelihood Methods in Nonparametric Statistical Inference and Confidence Region Estimation

Posted: May 21, 2009

• Owen Johnson
• Scarlett Martin
• Jacob Miller

Abstract

Empirical likelihood methods have emerged as a powerful nonparametric approach to statistical inference since their introduction by Owen in 1988. These methods combine the flexibility of nonparametric procedures with the efficiency of likelihood-based inference, allowing for the construction of confidence regions without requiring stringent distributional assumptions. The fundamental idea behind empirical likelihood is to replace the parametric likelihood function with a nonparametric analog that maximizes the likelihood subject to moment conditions or other estimating equations. This approach has found applications across various domains, including econometrics, biostatistics, and engineering. Despite their theoretical appeal and practical success in low-dimensional settings, empirical likelihood methods face significant challenges in high-dimensional problems. As the dimension of the parameter space increases, the empirical likelihood function can become degenerate, leading to confidence regions that are either too conservative or fail to maintain nominal coverage probabilities. This phenomenon, known as the curse of dimensionality, has limited the applicability of empirical likelihood methods in modern statistical problems where high-dimensional data are increasingly common. In this paper, we address these limitations through a novel framework that extends empirical likelihood methods to high-dimensional settings. Our approach combines information-theoretic principles with geometric optimization techniques, resulting in a methodology that preserves the desirable properties of empirical likelihood while overcoming the challenges associated with high dimensionality. We introduce three key innovations: a regularized empirical likelihood that incorporates structural constraints, a geometric reparameterization that transforms the optimization problem, and connections to optimal transport theory that enable the construction of confidence regions with guaranteed coverage properties.