We propose a composite hypothesis test in the Neyman-Pearson setting where the null distribution is known and the alternative distribution belongs to a certain family of distributions. The proposed test interpolates between Hoeffding’s test and the likelihood ratio test and achieves the optimal error exponent tradeoff for every distribution in the family. In addition, the proposed test is shown to attain the type-I error probability prefactor of ${n^{\frac{{\bar d - 1}}{2}}}$, where $\bar d$ is the dimension of the family of distributions projected onto a relative entropy ball centered at the null distribution. This can be significantly smaller than the prefactor ${n^{\frac{{a - 2}}{2}}}$ achieved by the Hoeffding’s test where d is the dimension of the probability simplex. In addition, the proposed test achieves the optimal type-II error probability prefactor for every distribution in the family.

We study the problem of mismatched binary hypothesis testing between i.i.d. distributions. We analyze the tradeoff between the pairwise error probability exponents when the actual distributions generating the observation are different from the distributions used in the likelihood ratio test, sequential probability ratio test, and Hoeffding’s generalized likelihood ratio test in the composite setting. When the real distributions are within a small divergence ball of the test distributions, we find the deviation of the worst-case error exponent of each test with respect to the matched error exponent. In addition, we consider the case where an adversary tampers with the observation, again within a divergence ball of the observation type. We show that the tests are more sensitive to distribution mismatch than to adversarial observation tampering.

We study mismatched sequential hypothesis testing. We analyze the type-I and and type-II error exponents when the actual distributions generating the observation are different from those used in the test. We derive the worst-case error exponents when the actual distributions generating the data are within a relative entropy ball of the test distributions and show the error exponent sensitivity of the test for small relative entropy balls.