Asymptotically Optimal One-and Two-Sample Testing with Kernels

Shengyu Zhu, Biao Chen, Zhitang Chen, Pengfei Yang

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


We characterize the asymptotic performance of nonparametric one-and two-sample testing. The exponential decay rate or error exponent of the type-II error probability is used as the asymptotic performance metric, and an optimal test achieves the maximum rate subject to a constant level constraint on the type-I error probability. With Sanov's theorem, we derive a sufficient condition for one-sample tests to achieve the optimal error exponent in the universal setting, i.e., for any distribution defining the alternative hypothesis. We then show that two classes of Maximum Mean Discrepancy (MMD) based tests attain the optimal type-II error exponent on Rd, while the quadratic-time Kernel Stein Discrepancy (KSD) based tests achieve this optimality with an asymptotic level constraint. For general two-sample testing, however, Sanov's theorem is insufficient to obtain a similar sufficient condition. We proceed to establish an extended version of Sanov's theorem and derive an exact error exponent for the quadratic-time MMD based two-sample tests. The obtained error exponent is further shown to be optimal among all two-sample tests satisfying a given level constraint. Our work hence provides an achievability result for optimal nonparametric one-and two-sample testing in the universal setting. Application to off-line change detection and related issues are also discussed.

Original languageEnglish (US)
Article number9354188
Pages (from-to)2074-2092
Number of pages19
JournalIEEE Transactions on Information Theory
Issue number4
StatePublished - Apr 2021


  • Universal hypothesis testing
  • error exponent
  • kernel Stein discrepancy (KSD)
  • large deviations
  • maximum mean discrepancy (MMD)

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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