  We investigate two closely related nonparametric hypothesis testing problems.
In the first problem (i.e., the existence problem), we test whether a testing
data stream is generated by one of a set of composite distributions. In the
second problem (i.e., the association problem), we test which one of the
multiple distributions generates a testing data stream. We assume that some
distributions in the set are unknown with only training sequences generated by
the corresponding distributions are available. For both problems, we construct
the generalized likelihood (GL) tests, and characterize the error exponents of
the maximum error probabilities. For the existence problem, we show that the
error exponent is mainly captured by the Chernoff information between the set
of composite distributions and alternative distributions. For the association
problem, we show that the error exponent is captured by the minimum Chernoff
information between each pair of distributions as well as the KL divergences
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 rate.
