  The achievable error-exponent pairs for the type I and type II errors are
characterized in a hypothesis testing setup where the observation consists of
independent and identically distributed samples from either a known joint
probability distribution or an unknown product distribution. The empirical
mutual information test, the Hoeffding test, and the generalized
likelihood-ratio test are all shown to be asymptotically optimal. An expression
based on a Renyi measure of dependence is shown to be the Fenchel biconjugate
of the error-exponent function obtained by fixing one error exponent and
optimizing the other. An example is provided where the error-exponent function
is not convex and thus not equal to its Fenchel biconjugate.
