On the Second-Order Asymptotics of the Hoeffding Test and Other Divergence Tests

Consider a composite hypothesis testing problem where the test has access to the null hypothesis P but not to the alternative hypothesis Q. The generalized likelihood-ratio test (GLRT) for this problem is the 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 paper proposes a generalization of the Hoeffding test, termed divergence test, for which the KL divergence is replaced by an arbitrary divergence. For this test, the first and second-order terms of the type-II error probability for a fixed type-I error probability are characterized and compared with the error terms of the Neyman-Pearson test, which is the optimal test when both P and Q are known. It is demonstrated that, irrespective of the divergence, divergence tests achieve the first-order term of the Neyman-Pearson test. 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. This implies that the GLRT may have a second-order asymptotic performance that is strictly suboptimal.