Abstract
We investigate a nonparametric hypothesis testing problem, in which we assume a testing data stream is generated by one of a set of distributions (hypotheses), and the goal is to test which one of the multiple distributions generates the testing data stream, i.e., which hypothesis occurs. We assume that some distributions in the set are unknown with only training sequences generated by the corresponding distributions are given. We construct the generalized likelihood (GL) test, and characterize the error exponent of the maximum error probability. We show that the error exponent is captured by the Chernoff distance between each pair of distributions as well as the KL divergence between the approximated distributions (via training sequences) and the true distributions. We also show that the ratio between the lengths of training and testing sequences plays an important role in determining the error decay behavior.
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