  We consider a problem of simple hypothesis testing using a randomized test
via a tunable loss function proposed by Liao \textit{et al}. In this problem,
we derive results that correspond to the Neyman--Pearson lemma, the
Chernoff--Stein lemma, and the Chernoff-information in the classical hypothesis
testing problem. Specifically, we prove that the optimal error exponent of our
problem in the Neyman--Pearson's setting is consistent with the classical
result. Moreover, we provide lower bounds of the optimal Bayesian error
exponent.
