Assessing goodness of fit to a given distribution plays an important role in computational statistics. The probability integral transformation (PIT) can be used to convert the question of whether a given sample originates from a reference distribution into a problem of testing for uniformity. We present new simulation- and optimization-based methods to obtain simultaneous confidence bands for the whole empirical cumulative distribution function (ECDF) of the PIT values under the assumption of uniformity. Simultaneous confidence bands correspond to such confidence intervals at each point that jointly satisfy a desired coverage. These methods can also be applied in cases where the reference distribution is represented only by a finite sample, which is useful, for example, for simulation-based calibration. The confidence bands provide an intuitive ECDF-based graphical test for uniformity, which also provides useful information on the quality of the discrepancy. We further extend the simulation and optimization methods to determine simultaneous confidence bands for testing whether multiple samples come from the same underlying distribution. This multiple sample comparison test is useful, for example, as a complementary diagnostic in multi-chain Markov chain Monte Carlo (MCMC) convergence diagnostics, where most currently used convergence diagnostics provide a single diagnostic value, but do not usually offer insight into the nature of the deviation. We provide numerical experiments to assess the properties of the tests using both simulated and real-world data and give recommendations on their practical application in computational statistics workflows.

评估给定分布的拟合优度在计算统计中起着重要作用。概率积分变换(PIT)可用于将给定样本是否源自参考分布的问题转换为均匀性检验问题。我们提出了新的基于模拟和优化的方法来获得整个经验累积分布函数的同时置信带(ECDF) 假设均匀性下的 PIT 值。同时置信带对应于在每个点处共同满足所需覆盖的置信区间。这些方法也可以应用于参考分布仅由有限样本表示的情况，这对于例如基于模拟的校准很有用。置信带提供了直观的基于 ECDF 的均匀性图形测试，它还提供了有关差异质量的有用信息。我们进一步扩展了模拟和优化方法，以确定同时置信带，以测试多个样本是否来自相同的基础分布。这种多样本比较测试很有用，例如，作为多链马尔可夫链蒙特卡罗 (MCMC) 收敛诊断中的补充诊断，其中大多数当前使用的收敛诊断提供单一诊断值，但通常不提供对偏差性质的洞察。我们提供数值实验来评估使用模拟和真实世界数据的测试特性，并就它们在计算统计工作流程中的实际应用提供建议。