We present methodology for constructing pointwise confidence intervals for the cumulative distribution function and the quantiles of mixing distributions on the unit interval from binomial mixture distribution samples. No assumptions are made on the shape of the mixing distribution. The confidence intervals are constructed by inverting exact tests of composite null hypotheses regarding the mixing distribution. Our method may be applied to any deconvolution approach that produces test statistics whose distribution is stochastically monotone for stochastic increase of the mixing distribution. We propose a hierarchical Bayes approach, which uses finite Polya Trees for modelling the mixing distribution, that provides stable and accurate deconvolution estimates without the need for additional tuning parameters. Our main technical result establishes the stochastic monotonicity property of the test statistics produced by the hierarchical Bayes approach. Leveraging the need for the stochastic monotonicity property, we explicitly derive the smallest asymptotic confidence intervals that may be constructed using our methodology. Raising the question whether it is possible to construct smaller confidence intervals for the mixing distribution without making parametric assumptions on its shape.

翻译：本文提出了一种方法，用于从二项混合分布样本构建单位区间上混合分布的累积分布函数和分位数的逐点置信区间。该方法不假设混合分布的形状，通过反转关于混合分布的复合零假设的精确检验来构建置信区间。我们的方法可应用于任何反卷积方法，只要其产生的检验统计量在混合分布随机递增时具有随机单调性。我们提出了一种层次贝叶斯方法，使用有限波利亚树对混合分布进行建模，该方法无需额外调参即可获得稳定且准确的反卷积估计。我们的主要技术结果是建立了层次贝叶斯方法产生的检验统计量的随机单调性。基于随机单调性的需求，我们明确推导了使用该方法可构建的最小渐近置信区间。这进而引出一个问题：是否有可能在不对其形状做参数假设的情况下，为混合分布构建更小的置信区间。