We analyzed the effect of the deviation of the exact distribution of the p-values from the uniform distribution on the Kolmogorov-Smirnov (K-S) test that was implemented as the second-level randomness test. We derived an inequality that provides an upper bound on the expected value of the K-S test statistic when the distribution of the null hypothesis differs from the exact distribution. Furthermore, we proposed a second-level test based on the two-sample K-S test with an ideal empirical distribution as a candidate for improvement.

翻译：我们分析了作为二级随机性测试而实施的Kolmogorov-Smirnov(K-S)测试中,P值的准确分布偏离了统一分布的精确分布的影响。当无效假设的分布与确切分布不同时,我们得出的不平等为K-S测试统计的预期值提供了上限。此外,我们提议基于两个样本的K-S测试进行二级测试,理想的经验分布作为改进的候选条件。