A nonparametric anomalous hypothesis testing problem is investigated, inwhich there are totally n sequences with s anomalous sequences to be detected.Each typical sequence contains m independent and identically distributed(i.i.d.) samples drawn from a distribution p, whereas each anomalous sequencecontains m i.i.d. samples drawn from a distribution q that is distinct from p.The distributions p and q are assumed to be unknown in advance.Distribution-free tests are constructed using maximum mean discrepancy as themetric, which is based on mean embeddings of distributions into a reproducingkernel Hilbert space. The probability of error is bounded as a function of thesample size m, the number s of anomalous sequences and the number n ofsequences. It is then shown that with s known, the constructed test isexponentially consistent if m is greater than a constant factor of log n, forany p and q, whereas with s unknown, m should has an order strictly greaterthan log n. Furthermore, it is shown that no test can be consistent forarbitrary p and q if m is less than a constant factor of log n, thus theorder-level optimality of the proposed test is established. Numerical resultsare provided to demonstrate that our tests outperform (or perform as well as)the tests based on other competitive approaches under various cases.
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