  In this paper we revisit the binary hypothesis testing problem with one-sided
compression. Specifically we assume that the distribution in the null
hypothesis is a mixture distribution of iid components. The distribution under
the alternative hypothesis is a mixture of products of either iid distributions
or finite order Markov distributions with stationary transition kernels. The
problem is studied under the Neyman-Pearson framework in which our main
interest is the maximum error exponent of the second type of error. We derive
the optimal achievable error exponent and under a further sufficient condition
establish the maximum $\epsilon$-achievable error exponent. It is shown that to
obtain the latter, the study of the exponentially strong converse is needed.
Using a simple code transfer argument we also establish new results for the
Wyner-Ahlswede-K{\"o}rner problem in which the source distribution is a mixture
of iid components.
