Exact null distributions of goodness-of-fit test statistics are generally challenging to obtain in tractable forms. Practitioners are therefore usually obliged to rely on asymptotic null distributions or Monte Carlo methods, either in the form of a lookup table or carried out on demand, to apply a goodness-of-fit test. There exist simple and useful transformations of several classic goodness-of-fit test statistics that stabilize their exact-$n$ critical values for varying sample sizes $n$. However, detail on the accuracy of these and subsequent transformations in yielding exact $p$-values, or even deep understanding on the derivation of several transformations, is still scarce nowadays. The latter stabilization approach is explained and automated to (i) expand its scope of applicability and (ii) yield upper-tail exact $p$-values, as opposed to exact critical values for fixed significance levels. Improvements on the stabilization accuracy of the exact null distributions of the Kolmogorov-Smirnov, Cram\'er-von Mises, Anderson-Darling, Kuiper, and Watson test statistics are shown. In addition, a parameter-dependent exact-$n$ stabilization for several novel statistics for testing uniformity on the hypersphere of arbitrary dimension is provided. A data application in astronomy illustrates the benefits of the advocated stabilization for quickly analyzing small-to-moderate sequentially-measured samples.

翻译：拟合优度检验统计量的精确零分布通常难以以易处理形式获得。因此，实践者通常不得不依赖渐近零分布或蒙特卡洛方法（以查找表形式或按需执行）来应用拟合优度检验。存在几种经典拟合优度检验统计量的简单有效变换，能够稳定其在不同样本量 $n$ 下的精确 $n$ 临界值。然而，关于这些变换及其后续变换在生成精确 $p$ 值方面的精度细节，甚至对若干变换推导过程的深入理解，至今仍较为匮乏。本文对后一种稳定性方法进行了解释并实现了自动化，以（i）扩展其适用范围，并（ii）生成上尾精确 $p$ 值，而非固定显著性水平下的精确临界值。展示了 Kolmogorov-Smirnov、Cramér-von Mises、Anderson-Darling、Kuiper 和 Watson 检验统计量在精确零分布稳定性精度上的改进。此外，对于任意维度超球面上均匀性检验的几种新型统计量，提供了参数相关的精确 $n$ 稳定性方法。一项天文学数据应用案例验证了所倡导的稳定性方法在快速分析小到中等规模顺序测量样本中的优势。