A HigherOrder KolmogorovSmirnov Test
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Proceedings of Machine Learning Research, PMLR 89:26212630, 2019.
Abstract
We present an extension of the KolmogorovSmirnov (KS) twosample test, which can be more sensitive to differences in the tails. Our test statistic is an integral probability metric (IPM) defined over a higherorder 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 $(k \le 5)$, we develop a lineartime algorithm for computing our higherorder KS test statistic; for all others $(k \ge 6)$, we give a nearly lineartime approximation. We derive the asymptotic null distribution for our test, and show that our nearly lineartime approximation shares the same asymptotic null. Lastly, we complement our theory with numerical studies.
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