Consider a binary statistical hypothesis testing problem, where $n$ independent and identically distributed random variables $Z^n$ are either distributed according to the null hypothesis $P$ or the alternative hypothesis $Q$, and only $P$ is known. A well-known test that is suitable for this case is the so-called Hoeffding test, which accepts $P$ if the Kullback-Leibler (KL) divergence between the empirical distribution of $Z^n$ and $P$ is below some threshold. This work characterizes the first and second-order terms of the type-II error probability for a fixed type-I error probability for the Hoeffding test as well as for divergence tests, where the KL divergence is replaced by a general divergence. It is demonstrated that, irrespective of the divergence, divergence tests achieve the first-order term of the Neyman-Pearson test, which is the optimal test when both $P$ and $Q$ are known. In contrast, the second-order term of divergence tests is strictly worse than that of the Neyman-Pearson test. It is further demonstrated that divergence tests with an invariant divergence achieve the same second-order term as the Hoeffding test, but divergence tests with a non-invariant divergence may outperform the Hoeffding test for some alternative hypotheses $Q$. Potentially, this behavior could be exploited by a composite hypothesis test with partial knowledge of the alternative hypothesis $Q$ by tailoring the divergence of the divergence test to the set of possible alternative hypotheses.

翻译：考虑一个二元统计假设检验问题，其中$n$个独立同分布随机变量$Z^n$要么服从原假设$P$，要么服从备择假设$Q$，且仅已知$P$。适用于该情形的一个著名检验是所谓的Hoeffding检验，该检验在$Z^n$的经验分布与$P$之间的Kullback-Leibler (KL)散度低于某个阈值时接受$P$。本研究刻画了在固定第一类错误概率下，Hoeffding检验以及将KL散度替换为一般散度的散度检验的第二类错误概率的一阶项和二阶项。结果表明，无论使用何种散度，散度检验都能达到Neyman-Pearson检验（当$P$和$Q$均已知时的最优检验）的一阶项。相比之下，散度检验的二阶项严格劣于Neyman-Pearson检验。进一步证明，具有不变散度的散度检验能达到与Hoeffding检验相同的二阶项，但具有非不变散度的散度检验可能在某些备择假设$Q$下优于Hoeffding检验。这一特性可被部分已知备择假设$Q$的复合假设检验所利用，通过针对可能的备择假设集合定制散度检验的散度来获得优势。