Comb Test: Histogram Uniformity Testing Based on Discrete Total Variation

Abstract

Histogram uniformity testing is a common statistical task usually performed using Pearson's chi-square test. This paper proposes a new test based on the discrete total variation that is easy to compute and, for comb-like (alternating) deviations, achieves up to 67% higher statistical power than Pearson's chi-square test, making it a complement to standard tests. The exact null distribution is computed via dynamic programming, and a gamma approximation with Monte Carlo estimation extends the test to arbitrarily large sample sizes. Experiments on simulated ADC alternating differential nonlinearity and on rounding bias detection in scientific data confirm the claims. The Python source code and precomputed data are available at https://github.com/DiscreteTotalVariation/CombTest.