Estimation in moderately misspecified models

2026-03-25

AI Summary

This study addresses the fundamental trade-off between bias in a narrow model and variance in a wider model under moderate misspecification—where the true data-generating process includes one additional parameter beyond the fitted narrow model. The authors introduce the concept of a “tolerance radius” to quantify the range of misspecification within which the narrow model yields superior performance. Building on large-sample theory, likelihood-based estimation, and bias–variance decomposition, they develop a novel estimator that achieves robustness and efficiency across both model classes. Theoretical analysis and extensive numerical experiments across multiple model settings demonstrate that the proposed estimator significantly improves estimation accuracy within the tolerance radius, offering a principled balance between robustness to misspecification and statistical efficiency.

Technology Category

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

Application Category

• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Suppose data are fitted to some parametric model but that the true model happens to be one with an additional parameter. When a parameter is to be estimated one can use likelihood estimation in the wider model or in the narrow model. Including the extra parameter in the model means less bias but larger sampling variability. Two basic questions are addressed in this article. (i) Just how much misspecification can the narrow model tolerate? In the context of a large-sample moderate misspecification framework we find a surprisingly simple, sharp, and general answer. There is effectively a `tolerance radius' around a given narrow model, inside of which narrow estimation is more precise than wide estimation for all estimands. This is computed in a selection of examples that also demonstrate the degree of robustness of important standard methods against moderate incorrectness of the model under which they are optimal. (ii) Are there other estimators that work well both under narrow and wide circumstances? We discuss several possibilities and propose some new procedures. All methods are compared in a broad large-sample performance study.

Problem

• model misspecification
• parameter estimation
• narrow model
• wide model
• estimator robustness

Innovation

• moderate misspecification
• tolerance radius
• narrow vs. wide models
• robust estimation
• likelihood estimation