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：本研究重新探讨了单样本与双样本检验问题：即在一个或两个分布未知情况下的二元假设检验。针对单样本检验，我们为霍夫丁似然比检验的渐近最优性提供了更为简明的证明，该检验等价于经验分布与名义分布之间相对熵的阈值检验。新证明给出了直观的阐释，并自然推广至双样本检验场景。我们证明霍夫丁检验的类似形式——即两个经验分布之间相对熵的阈值检验——同样具有渐近最优性。此外，我们还获得了双样本检验的强逆定理。