On the attainment of the Wasserstein--Cramer--Rao lower bound
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908499843219456 |
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| author | Nishimori, Hayato Matsuda, Takeru |
| author_facet | Nishimori, Hayato Matsuda, Takeru |
| contents | Recently, a Wasserstein analogue of the Cramer--Rao inequality has been developed using the Wasserstein information matrix (Otto metric). This inequality provides a lower bound on the Wasserstein variance of an estimator, which quantifies its robustness against additive noise. In this study, we investigate conditions for an estimator to attain the Wasserstein--Cramer--Rao lower bound (asymptotically), which we call the (asymptotic) Wasserstein efficiency. We show a condition under which Wasserstein efficient estimators exist for one-parameter statistical models. This condition corresponds to a recently proposed Wasserstein analogue of one-parameter exponential families (e-geodesics). We also show that the Wasserstein estimator, a Wasserstein analogue of the maximum likelihood estimator based on the Wasserstein score function, is asymptotically Wasserstein efficient in location-scale families. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_12732 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the attainment of the Wasserstein--Cramer--Rao lower bound Nishimori, Hayato Matsuda, Takeru Statistics Theory Machine Learning Recently, a Wasserstein analogue of the Cramer--Rao inequality has been developed using the Wasserstein information matrix (Otto metric). This inequality provides a lower bound on the Wasserstein variance of an estimator, which quantifies its robustness against additive noise. In this study, we investigate conditions for an estimator to attain the Wasserstein--Cramer--Rao lower bound (asymptotically), which we call the (asymptotic) Wasserstein efficiency. We show a condition under which Wasserstein efficient estimators exist for one-parameter statistical models. This condition corresponds to a recently proposed Wasserstein analogue of one-parameter exponential families (e-geodesics). We also show that the Wasserstein estimator, a Wasserstein analogue of the maximum likelihood estimator based on the Wasserstein score function, is asymptotically Wasserstein efficient in location-scale families. |
| title | On the attainment of the Wasserstein--Cramer--Rao lower bound |
| topic | Statistics Theory Machine Learning |
| url | https://arxiv.org/abs/2506.12732 |