Second-level randomness test based on the Kolmogorov-Smirnov test

TL;DR

This paper investigates how deviations in p-value distributions affect the Kolmogorov-Smirnov test used for randomness testing, deriving bounds and proposing an improved second-level test based on the two-sample K-S test.

Contribution

It derives an inequality for the K-S test statistic under distribution deviations and proposes an enhanced second-level test using the two-sample K-S test.

Findings

Derived an upper bound inequality for the K-S test statistic.

Proposed an improved second-level randomness test.

Analyzed the impact of distribution deviations on the K-S test.

Abstract

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.