A Higher-Order Kolmogorov-Smirnov Test
This is an incremental improvement for statistical testing, specifically enhancing sensitivity to tail differences in two-sample comparisons.
The paper tackles the problem of making the Kolmogorov-Smirnov two-sample test more sensitive to differences in distribution tails by extending it to a higher-order integral probability metric, with results including linear-time algorithms for small orders and asymptotic null distribution derivations.
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 ($k \leq 5$), we develop a linear-time algorithm for computing our higher-order KS test statistic; for all others ($k \geq 6$), 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.