We prove lower bounds on the error of any estimator for the mean of a real probability distribution under the knowledge that the distribution belongs to a given set. We apply these lower bounds both to parametric and nonparametric estimation. In the nonparametric case, we apply our results to the question of sub-Gaussian estimation for distributions with finite variance to obtain new lower bounds in the small error probability regime, and present an optimal estimator in that regime. In the (semi-)parametric case, we use the Fisher information to provide distribution-dependent lower bounds that are constant-tight asymptotically, of order $\sqrt{2\log(1/\delta)/(nI)}$ where $I$ is the Fisher information of the distribution. We use known minimizers of the Fisher information on some nonparametric set of distributions to give lower bounds in cases such as corrupted distributions, or bounded/semi-bounded distributions.

翻译：我们证明了在已知分布属于给定集合的情况下，对任一实数概率分布均值估计误差的下界。我们将这些下界应用于参数估计和非参数估计。在非参数情形下，我们将结果应用于有限方差分布的子高斯估计问题，在小误差概率区域获得了新的下界，并给出了该区域的最优估计量。在（半）参数情形下，我们利用Fisher信息给出了分布依赖的下界，该下界渐近地具有常数紧致性，形式为$\sqrt{2\log(1/\delta)/(nI)}$，其中$I$为分布的Fisher信息。我们利用某些非参数分布集合上已知的Fisher信息最小化量，在污染分布或有界/半有界分布等情形下给出了下界。