Exact Confidence Intervals for the Mixing Distribution from Binomial Mixture Distribution Samples

We present a 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 about 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 a stochastic increase of the mixing distribution. We propose a hierarchical Bayes approach, which uses finite Pólya Trees to model the mixing distribution, that provides stable and accurate deconvolution estimates without 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. This raises the question of whether it is possible to construct smaller confidence intervals for the mixing distribution without making parametric assumptions about its shape. Supplementary materials for this article are available online.