In this work, we revisit the one- and two-sample testing problems: binary hypothesis testing in which one or both distributions are unknown. For the one-sample test, we provide a more streamlined proof of the asymptotic optimality of Hoeffding's likelihood ratio test, which is equivalent to the threshold test of the relative entropy between the empirical distribution and the nominal distribution. The new proof offers an intuitive interpretation and naturally extends to the two-sample test where we show that a similar form of Hoeffding's test, namely a threshold test of the relative entropy between the two empirical distributions is also asymptotically optimal. A strong converse for the two-sample test is also obtained.

翻译：本文重新审视了单样本和双样本的检验问题：即一个或两个分布均未知的二元假设检验。对于单样本检验，我们提供了Hoeffding似然比检验渐近最优性的一种更简洁的证明，该检验等价于对经验分布与名义分布间相对熵的阈值检验。这一新证明提供了直观的解释，并自然地推广至双样本检验——我们证明了类似形式的Hoeffding检验（即对两个经验分布间相对熵的阈值检验）同样具有渐近最优性。此外，还得到了双样本检验的一个强逆结论。