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Semiparametrics via parametrics and contiguity (2501.09483v1)

Published 16 Jan 2025 in math.ST, econ.EM, stat.ME, and stat.TH

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.

Summary

  • The paper establishes that parametric approximations under contiguity yield asymptotically normal and efficient MLE estimates in semiparametric models.
  • It introduces a robust framework using contiguity and DQM to overcome challenges from infinite-dimensional nuisance parameters.
  • Applications to partially linear and Cox regression models demonstrate the method's practical impact in econometrics and biostatistics.

Semiparametrics via Parametrics and Contiguity

The paper of semiparametric models is pivotal in statistical inference due to the flexibility they offer by incorporating both parametric and nonparametric components. The paper "Semiparametrics via Parametrics and Contiguity" by Adam Lee, Emil A. Stoltenberg, and Per A. Mykland discusses a novel approach to infer the parametric components of semiparametric models by leveraging parametric approximations.

Overview

The core challenge addressed in the paper is the nontrivial task of deriving estimators for the parametric components in semiparametric models amidst infinite-dimensional nuisance parameters. Traditional methods often stumble due to the intrinsic complexities associated with infinite dimensions. The paper proposes an innovative technique: approximate the infinite-dimensional component with a growing sequence of parametric models. This approximation permits the use of maximum likelihood estimation (MLE) to infer the parametric component under the assumption of mutual contiguity between the parametric approximations and the true semiparametric model.

Key Contributions

  1. Asymptotic Efficiency and Normality: The authors rigorously establish conditions under which the sequence of parametric model estimators converges in distribution to an asymptotically normal and efficient estimator in the semiparametric context. By ensuring that the parametric models are asymptotically equivalent to the semiparametric model, they demonstrate that the asymptotic properties of MLE can be effectively transferred.
  2. Technical Framework: The concept of contiguity and differentiability in quadratic mean (DQM) is used to bridge the parametric and semiparametric domains. The paper delineates conditions under which the likelihood ratios and the score functions of the parametric approximations converge in the requisite manner.
  3. Applications to Canonical Models: The theory is illustrated using the partially linear regression model and the Cox regression model. The paper provides detailed derivations of the semiparametric efficient information matrices and score functions, showcasing the applicability and robustness of their method in handling well-known semiparametric models.
  4. Sieved Profile Likelihood Approach: An alternative perspective is introduced whereby the profile likelihood method is adapted to the sieve framework. The concept of sieved models allows for handling infinite-dimensional parameters more tractably within a finite-dimensional framework.

Implications

The implications are significant in both theoretical and applied statistics. The ability to handle the parametric component of semiparametric models using classical MLE provides a more straightforward and computationally feasible approach. This could advance the development of more efficient algorithms and software for statistical inference in semiparametric contexts.

Moreover, the methodological innovation enhances the interpretability of models where the parametric portion is of primary interest, while still accounting for complex nonparametric influences. These insights are particularly valuable in econometrics and biostatistics, where semiparametric models are frequently utilized due to their adaptability and broader applicability.

Future Directions

Future work could explore extending these techniques to more complex structures, such as models with random effects or other dependent data structures not covered by the current framework. Additionally, computational advancements in handling large parametric spaces efficiently could be addressed, potentially involving machine learning algorithms that can manage high-dimensional approximations more effectively.

In conclusion, this paper makes substantial contributions to semiparametric statistical theory by integrating parametric approximations through contiguity and developing a rigorous framework for their application. This approach not only simplifies certain aspects of semiparametric inference but also promises to broaden the toolkit available to statisticians tackling complex models where traditional methods falter.

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