We address the problem of producing a lower bound for the mean of a discrete probability distribution, with known support over a finite set of real numbers, from an iid sample of that distribution. Up to a constant, this is equivalent to bounding the mean of a multinomial distribution (with known support) from a sample of that distribution. Our main contribution is to characterize the complete set of admissible bound functions for any sample space, and to show that certain previously published bounds are admissible. We prove that the solution to each one of a set of simple-to-state optimization problems yields such an admissible bound. Single examples of such bounds, such as the trinomial bound by Miratrix and Stark [2009] have been previously published, but without an analysis of admissibility, and without a discussion of the full set of alternative admissible bounds. In addition to a variety of results about admissible bounds, we prove the non-existence of optimal bounds for sample spaces with supports of size greater than 1 and samples sizes greater than 1.

翻译：本文研究从已知支撑集为有限实数集的离散概率分布的独立同分布样本中，构造该分布均值的下界问题。在常数意义下，该问题等价于从多项分布样本中推断其均值（已知支撑集）。我们的主要贡献在于刻画了任意样本空间上所有可容许界函数的完备集合，并证明某些已发表界具有可容许性。我们证明，通过求解一组表述简洁的优化问题即可得到此类可容许界。此前已有此类界的个别实例发表（如Miratrix和Stark[2009]提出的三项界），但未进行可容许性分析，亦未讨论可容许界的完备集合。除关于可容许界的多项结论外，我们还证明了当支撑集规模大于1且样本量大于1时，最优界不存在。