On the attainment of the Wasserstein--Cramer--Rao lower bound

• URL: http://arxiv.org/abs/2506.12732v2
• Date: Tue, 17 Jun 2025 15:04:48 GMT
• Title: On the attainment of the Wasserstein--Cramer--Rao lower bound
• Authors: Hayato Nishimori, Takeru Matsuda,
• Abstract summary: A Wasserstein analogue of the Cramer--Rao inequality has been developed using the Wasserstein information matrix (Otto metric)<n>This inequality provides a lower bound on the Wasserstein variance of an estimator, which quantifies its robustness against additive noise.<n>We investigate conditions for an estimator to attain the Wasserstein--Cramer--Rao lower bound, which we call the (asymptotic) Wasserstein efficiency.
• Score: 4.48890356952206
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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