# A Higher-Order Kolmogorov-Smirnov Test

arXiv: https://arxiv.org/abs/1903.10083 · HTML: https://mlsanity.com/paper/1903.10083/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Mar 2019.

## Built on by (most cited first, 4 of 4)

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- [Integral Probability Metrics Meet Neural Networks: The Radon-Kolmogorov-Smirnov Test](https://mlsanity.com/paper/2309.02422/) (2023)
- [Efficient and Stable Multi-Dimensional Kolmogorov-Smirnov Distance](https://mlsanity.com/paper/2504.11299/) (2025)
- [Data Fusion with Distributional Equivalence Test-then-pool](https://mlsanity.com/paper/2603.11867/) (2026)

## Abstract (arXiv)

We present an extension of the Kolmogorov-Smirnov (KS) two-sample test, which can be more sensitive to differences in the tails. Our test statistic is an integral probability metric (IPM) defined over a higher-order total variation ball, recovering the original KS test as its simplest case. We give an exact representer result for our IPM, which generalizes the fact that the original KS test statistic can be expressed in equivalent variational and CDF forms. For small enough orders (k ≤ 5), we develop a linear-time algorithm for computing our higher-order KS test statistic; for all others (k ≥ 6), we give a nearly linear-time approximation. We derive the asymptotic null distribution for our test, and show that our nearly linear-time approximation shares the same asymptotic null. Lastly, we complement our theory with numerical studies.