标题： 多项式参数的小样本置信区域 显示英文标题

标题： Confidence regions for the multinomial parameter with small sample size

摘要： 考虑n个独立同分布的实验结果，该实验有d>1种可能的结果，这相当于对多项分布M(n,p)的一个观察，其中p是{1,...,d}上的未知离散分布。 在许多应用中，当n较小时构建p的置信区域至关重要。 这个具体的挑战性问题有着悠久的历史。 众所周知，当n较小时，基于渐近统计量构建的置信区域覆盖效果不佳。 另一方面，大多数提供非渐近区域且具有受控覆盖的方法仅限于二项情况d=2。 在本工作中，我们提出了一种适用于任何d>1的新方法。 该方法提供了具有受控覆盖和较小体积的置信区域，并且包括对与似然水平集相关的“覆盖集合”进行反演。 当d/n趋于无穷时的行为仍然是一个有趣但超出本文范围的开放问题。 显示英文摘要

摘要： Consider the observation of n iid realizations of an experiment with d>1 possible outcomes, which corresponds to a single observation of a multinomial distribution M(n,p) where p is an unknown discrete distribution on {1,...,d}. In many applications, the construction of a confidence region for p when n is small is crucial. This concrete challenging problem has a long history. It is well known that the confidence regions built from asymptotic statistics do not have good coverage when n is small. On the other hand, most available methods providing non-asymptotic regions with controlled coverage are limited to the binomial case d=2. In the present work, we propose a new method valid for any d>1. This method provides confidence regions with controlled coverage and small volume, and consists of the inversion of the "covering collection"' associated with level-sets of the likelihood. The behavior when d/n tends to infinity remains an interesting open problem beyond the scope of this work.