Composite Neyman-Pearson Hypothesis Testing with a Known Hypothesis

Abstract

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.