On the role of parameterization in models with a misspecified nuisance component

Area of Science:

• Statistics
• Econometrics
• Machine Learning

Background:

• Statistical inference often relies on correctly specified models, which can be restrictive.
• Maximum likelihood estimators (MLEs) are widely used but can be sensitive to model misspecification, particularly in nuisance parameters.
• Existing results for specific problems like matched-comparison and two-groups lack a unified theoretical framework.

Purpose of the Study:

• To identify general conditions under which maximum likelihood estimators are consistent, even with misspecified nuisance parameters.
• To unify and generalize existing results in matched-comparison and two-groups problems.
• To explore the role of parameter orthogonality in robust statistical inference.

Main Methods:

• Development of a general theoretical framework for analyzing the consistency of MLEs under model misspecification.
• Specialization of the general results to matched-comparison and two-groups settings.
• Investigation of generalized parameter orthogonality and its connection to Neyman orthogonality.

Main Results:

• A general condition is derived for the consistency of the maximum likelihood estimator (MLE) of a parameter of interest under arbitrary misspecification of nuisance parameters.
• A simplified and verifiable condition based on symmetric parameterization is established for matched-comparison and two-groups problems, unifying prior findings.
• The study highlights the significance of generalized parameter orthogonality and its relationship with Neyman orthogonality for robust inference.

Conclusions:

• The paper provides a robust framework for statistical inference, ensuring estimator consistency despite potential model misspecification.
• The findings offer a unified approach to problems involving matched-comparison and two-groups, simplifying existing conditions.
• The research underscores the importance of parameter orthogonality in developing reliable inferential methods beyond simple consistency.