[Paper Review] Estimation in moderately misspecified models

Nils Lid Hjort|ArXiv.org|Mar 25, 2026

Statistical Methods and InferenceMathematics7 references9 citations

TL;DR

The paper analyzes how much misspecification a narrow parametric model can tolerate and compares estimation in narrow versus wide models, deriving a sharp large-sample tolerance criterion and proposing compromise estimators.

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.

Motivation & Objective

• Quantify how much misspecification a narrow model can tolerate before wide-model estimation becomes preferable.
• Develop a large-sample framework to compare narrow and wide estimators under local misspecification.
• Provide a simple, general tolerance criterion based on information matrices that applies across common models.
• Propose and evaluate compromise estimators that perform well under both narrow and wide circumstances.
• Discuss connections to model selection criteria and Bayesian robustness.

Proposed method

• Set up a large-sample framework with a narrow model f(y, θ) and a wider model f(y, θ, γ) where γ deviates from γ0 by δ/√n.
• Derive the limiting distributions for the wide estimator (θ̂, γ̂) and the narrow estimator θ̂ narr, across local misspecifications (γ = γ0 + δ/√n).
• Compute the comparison metric n · MSE for the two estimators and obtain a general criterion δ^2 ≤ κ^2 for when the narrow estimator is preferable.
• Express κ^2 via the information matrix J wide and show independence from the specific estimand μ(θ, γ).
• Extend results to regression settings with covariates and to multiple example models.
• Relate findings to AIC and Schwarz criteria and discuss robustness and pre-test implications.

Experimental results

Research questions

• RQ1 How much misspecification in the wider model can the narrow model tolerate before narrow estimation becomes less precise?
• RQ2 Under local misspecification, when is the narrow-model estimator better than the wide-model estimator in terms of asymptotic mean squared error?
• RQ3 Can a simple, general tolerance radius κ^2 be computed from information matrices, applicable across common parametric models?
• RQ4 What are practical compromise estimators that perform well under both narrow and wide conditions, and how do they compare to standard methods?
• RQ5 How do Akaike and Schwarz model selection criteria relate to the tolerance framework proposed?

Key findings

• A simple, sharp large-sample tolerance criterion is derived: the narrow estimator is preferable whenever δ^2 ≤ κ^2, with κ^2 computed from the wide-model information matrix evaluated at the narrow model.
• The tolerance criterion does not depend on the particular estimand μ, making it broadly applicable across examples.
• The paper demonstrates how to compute κ^2 from J wide (the information matrix of the wide model) evaluated at the narrow model, facilitating practical assessment of robustness.
• If the true model deviates modestly (borderline case), the probability of detecting misspecification at level 5% is around 17%, illustrating a practical boundary for model choice.
• Akaike’s information criterion and Schwarz’s criterion are analyzed in this framework, showing how their decisions relate to the narrow-vs-wide trade-off under misspecification.
• Extensions to regression with covariates and to more general departures are discussed, providing a broad applicability of the tolerance approach.