We study divergence-based tests for statistical hypothesis testing in settings where the underlying distributions are only partially known. First, we consider a one-sample binary hypothesis testing problem in which the observations are i.i.d. according to either a known distribution (P) or an unknown alternative (Q). The classical test for this problem is the Hoeffding test which accepts the null hypothesis when the Kullback–Leibler (KL) divergence between the empirical distribution and (P) is below a prescribed threshold. We generalize the Hoeffding test by replacing the KL divergence with an arbitrary divergence which we refer to as the divergence test. We characterise the first- and second-order asymptotics of the type-II error probability for a fixed type-I error probability. We show that, irrespective of the divergence, divergence tests achieve the optimal first-order exponent of the Neyman–Pearson test, while their second-order performance is generally inferior. Divergence tests based on invariant divergences attain the same second-order term as the Hoeffding test, whereas suitably chosen non-invariant divergences can outperform it for certain alternatives. We further extend this framework to two-sample testing, where one observes two independent sequences of i.i.d. random variables distributed according to (P_1) and (P_2), respectively, and wishes to determine whether (P_1=P_2) (null hypothesis) or (P_1 eq P_2) (alternative hypothesis). We generalise the Gutman test, which is designed for two-sample testing, by replacing the Jensen–Shannon divergence with an arbitrary divergence. We show that divergence tests achieve the optimal first-order exponent and that tests based on invariant divergences have the same second-order asymptotics as the Gutman test. These results provide a unified perspective on divergence-based universal hypothesis testing and establish connections between two-sample testing and robust goodness-of-fit testing.
Jithin Ravi received the M.Tech. and Ph.D. degrees in electrical engineering from Indian Institute of Technology Bombay, Mumbai, in 2017. He completed his first postdoctoral fellowship at the Tata Institute of Fundamental Research, Mumbai, in 2017, followed by a second post-doctoral position at the Signal Theory and Communications Department, Universidad Carlos III de Madrid, Leganes, Spain, from 2018 to 2021. He is currently an Assistant Professor with the Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology Kharagpur. His research interests include information theory, wireless communications, information-theoretic security, and statistical inference.