Comb Test: Histogram Uniformity Testing Based on Discrete Total Variation

2026-05-31

AI Summary

This study addresses the limited statistical power of the conventional Pearson chi-squared test under comb-like (alternating) deviations from uniformity. To overcome this limitation, the authors propose a histogram uniformity test based on discrete total variation. The method employs dynamic programming to compute the exact null distribution and integrates gamma approximation with Monte Carlo simulation to accommodate arbitrary sample sizes. Under comb-shaped alternatives, the proposed test achieves up to a 67% increase in statistical power compared to Pearson’s chi-squared test, substantially improving detection sensitivity for alternating nonlinearities and rounding biases. Empirical evaluations demonstrate its superior performance in detecting differential nonlinearity in simulated analog-to-digital converters and rounding artifacts in scientific data. The implementation code and datasets are publicly available.

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.

Problem

• histogram uniformity testing
• discrete total variation
• statistical power
• comb-like deviations
• chi-square test

Innovation

• discrete total variation
• histogram uniformity testing
• comb-like deviations
• statistical power
• dynamic programming