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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2412.08475 |
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| _version_ | 1866911099736031232 |
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| author | Vos, Paul W. |
| author_facet | Vos, Paul W. |
| contents | The James-Stein estimator's dominance over maximum likelihood in terms of mean square error (MSE) has been one of the most celebrated results in modern statistics, suggesting that biased estimators can systematically outperform unbiased ones. We argue that this conclusion stems from using an inappropriate assessment criterion. Through simple simulations, we demonstrate that while James-Stein achieves lower MSE, it produces concerning behavior in practice: hypothesis tests based on James-Stein can have power below the significance level, exhibit severe asymmetry, and lead to conclusions that practitioners would hesitate to report. Using $Λ$-information (Vos and Wu, 2025), a criterion that measures how effectively estimators distinguish between distributions, we show that maximum likelihood achieves full efficiency while James-Stein performs poorly precisely where MSE suggests superiority. Our analysis reveals that MSE's fundamental flaw--assessing estimators point-wise thereby missing important aspects of estimation--creates these paradoxes. By expanding from point estimators to generalized estimators (functions over the parameter space), we obtain a parameter-invariant framework that unifies estimation and testing. These insights suggest that the statistical community should reconsider not maximum likelihood theory, but rather our reliance on MSE for comparing estimators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08475 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rethinking Mean Square Error: Why Information is a Superior Assessment of Estimators Vos, Paul W. Statistics Theory The James-Stein estimator's dominance over maximum likelihood in terms of mean square error (MSE) has been one of the most celebrated results in modern statistics, suggesting that biased estimators can systematically outperform unbiased ones. We argue that this conclusion stems from using an inappropriate assessment criterion. Through simple simulations, we demonstrate that while James-Stein achieves lower MSE, it produces concerning behavior in practice: hypothesis tests based on James-Stein can have power below the significance level, exhibit severe asymmetry, and lead to conclusions that practitioners would hesitate to report. Using $Λ$-information (Vos and Wu, 2025), a criterion that measures how effectively estimators distinguish between distributions, we show that maximum likelihood achieves full efficiency while James-Stein performs poorly precisely where MSE suggests superiority. Our analysis reveals that MSE's fundamental flaw--assessing estimators point-wise thereby missing important aspects of estimation--creates these paradoxes. By expanding from point estimators to generalized estimators (functions over the parameter space), we obtain a parameter-invariant framework that unifies estimation and testing. These insights suggest that the statistical community should reconsider not maximum likelihood theory, but rather our reliance on MSE for comparing estimators. |
| title | Rethinking Mean Square Error: Why Information is a Superior Assessment of Estimators |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2412.08475 |