Hybrid combinations of parametric and empirical likelihoods

2026-02-20

AI Summary

This study addresses the challenge of achieving both estimation efficiency and robustness in the presence of potential parametric model misspecification. The authors propose a hybrid likelihood approach that combines parametric and empirical likelihoods through a carefully designed weighting scheme, yielding an estimation framework that balances efficiency with robustness. They introduce a novel formulation of the hybrid likelihood function and establish the asymptotic normality of the resulting estimator along with a Wilks-type theorem, ensuring reliable performance even under model misspecification. Theoretical analysis demonstrates that the associated likelihood ratio statistic converges to a standard chi-squared distribution, providing a solid foundation for statistical inference. Additionally, a data-driven strategy is developed for selecting the tuning parameter that governs the trade-off between efficiency and robustness.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

This paper develops a hybrid likelihood (HL) method based on a compromise between parametric and nonparametric likelihoods. Consider the setting of a parametric model for the distribution of an observation $Y$ with parameter $θ$. Suppose there is also an estimating function $m(\cdot,μ)$ identifying another parameter $μ$ via $E\,m(Y,μ)=0$, at the outset defined independently of the parametric model. To borrow strength from the parametric model while obtaining a degree of robustness from the empirical likelihood method, we formulate inference about $θ$ in terms of the hybrid likelihood function $H_n(θ)=L_n(θ)^{1-a}R_n(μ(θ))^a$. Here $a\in[0,1)$ represents the extent of the compromise, $L_n$ is the ordinary parametric likelihood for $θ$, $R_n$ is the empirical likelihood function, and $μ$ is considered through the lens of the parametric model. We establish asymptotic normality of the corresponding HL estimator and a version of the Wilks theorem. We also examine extensions of these results under misspecification of the parametric model, and propose methods for selecting the balance parameter $a$.

Problem

• hybrid likelihood
• parametric likelihood
• empirical likelihood
• robust inference
• model misspecification

Innovation

• hybrid likelihood
• empirical likelihood
• parametric model
• robustness
• model misspecification