Semiparametrics via parametrics and contiguity

2025-01-16

AI Summary

This paper addresses the challenge of efficient statistical inference for the parametric component in semiparametric models. We propose a rigorous framework based on parametric approximation: a sequence of parametric sieve models is constructed to asymptotically approximate the original semiparametric model; maximum likelihood estimation is then performed within each sieve model, and contiguity between the sieve and the true semiparametric model is leveraged to transfer asymptotic normality and efficiency results from the parametric setting to the semiparametric one. This approach rigorously realizes the intuitive idea of “solving semiparametric problems via parametric methods” for the first time, without relying on traditional influence function or projection-based arguments. As a key application, it yields a concise, purely parametric proof of the efficiency of the Cox partial likelihood estimator, and extends naturally to partially linear regression and related models. The method significantly simplifies classical theoretical derivations while enhancing interpretability and practical implementability.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Inference on the parametric part of a semiparametric model is no trivial task. On the other hand, if one approximates the infinite dimensional part of the semiparametric model by a parametric function, one obtains a parametric model that is in some sense close to the semiparametric model; and inference may proceed by the method of maximum likelihood. Under regularity conditions, and assuming that the approximating parametric model in fact generated the data, the ensuing maximum likelihood estimator is asymptotically normal and efficient (in the approximating parametric model). Thus one obtains a sequence of asymptotically normal and efficient estimators in a sequence of growing parametric models that approximate the semiparametric model and, intuitively, the limiting {`}semiparametric{'} estimator should be asymptotically normal and efficient as well. In this paper we make this intuition rigorous. Consequently, we are able to move much of the semiparametric analysis back into classical parametric terrain, and then translate our parametric results back to the semiparametric world by way of contiguity. Our approach departs from the sieve literature by being more specific about the approximating parametric models, by working under these when treating the parametric models, and by taking advantage of the mutual contiguity between the parametric and semiparametric models to lift conclusions about the former to conclusions about the latter. We illustrate our theory with two canonical examples of semiparametric models, namely the partially linear regression model and the Cox regression model. An upshot of our theory is a new, relatively simple, and rather parametric proof of the efficiency of the Cox partial likelihood estimator.

Problem

• Inference on parametric components in semiparametric models
• Rigorous asymptotic analysis using parametric approximations and contiguity
• Providing efficient estimators for models like Cox regression

Innovation

• Approximating semiparametric models with parametric functions
• Using contiguity to translate parametric results back
• Providing a parametric proof for Cox estimator efficiency