标题： 关于达到Wasserstein-Cramer-Rao下界的研究 显示英文标题

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

摘要： 近期，利用Wasserstein信息矩阵（Otto度量），开发出了一个Cramér-Rao不等式的Wasserstein类比。 这一不等式为估计量的Wasserstein方差提供了下界，量化了其对加性噪声的鲁棒性。 在本研究中，我们探讨了估计量达到Wasserstein-Cramér-Rao下界的条件（渐近情况下），我们将此称为（渐近）Wasserstein有效性。 我们展示了对于单参数统计模型，存在Wasserstein有效估计量的一个条件。 该条件对应于最近提出的单参数指数族的Wasserstein类比（e-测地线）。 我们还证明了基于Wasserstein得分函数的Wasserstein估计量，在位置尺度族中是渐近Wasserstein有效的。 显示英文摘要

摘要： 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.