Adam Lee, Emil A. Stoltenberg, Per A. Mykland
arXiv 16 Jan 2025 · Mathematics — Statistics Theory
arXiv:2501.09483 · PDF · DOI · OpenAlex · Extracted main text
Inference on the parametric part of a semiparametric model is no trivial task. 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, 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: we 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 conventional sieve literature by being more specific about the approximating parametric models, by working not only with but also under these when treating the parametric models, and by taking full advantage of the mutual contiguity that we require between the parametric and semiparametric models. 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.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Aad W van der Vaart (1998) Asymptotic Statistics | 1.000 | 10 | 5 | 100% |
| 2 | Peter J. Bickel, Chris. A. J. Klaassen, Ya'acov Ritov, and Jon A. We… (1998) Efficient and Adaptive Estimation for Semiparametric Models | 0.928 | 4 | 3 | 100% |
| 3 | Susan A Murphy and Aad W van der Vaart (2000) On profile likelihood | 0.874 | 9 | 2 | 100% |
| 4 | Per Kragh Andersen, rnulf Borgan, Richard D Gill, and Niels Keiding (1993) Statistical Models Based on Counting Processes | 0.843 | 3 | 3 | 100% |
| 5 | N. L. Hjort and D. B. Pollard (1993) Asymptotics for minimisers of convex processes | 0.811 | 4 | 2 | 100% |
| 6 | Lucien Le Cam (1986) Asymptotic Methods in Statistical Decision Theory | 0.644 | 4 | 1 | 100% |
| 7 | Per Kragh Andersen and Richard D Gill (1982) Cox's regression model for counting processes: A large sample study | 0.644 | 2 | 2 | 100% |
| 8 | Allan Gut (1992) The weak law of large numbers for arrays | 0.644 | 2 | 2 | 100% |
| 9 | Jean Jacod and Albert Shiryaev (2003) Limit Theorems for Stochastic Processes. Second Edition | 0.644 | 2 | 2 | 100% |
| 10 | Helmut Strasser Mathematical Theory of Statistics | 0.644 | 2 | 2 | 100% |
Showing the top 10 of 33 scored citations.