Asymptotically Optimal Tests for One- and Two-Sample Problems
For statisticians and machine learning researchers, this work offers a unified, intuitive proof of optimality for both one- and two-sample tests, though the results are largely theoretical and incremental.
The paper revisits one- and two-sample testing problems, providing a streamlined proof of asymptotic optimality for Hoeffding's likelihood ratio test in the one-sample case and extending it to show that a similar test based on relative entropy between empirical distributions is asymptotically optimal for the two-sample case, including a strong converse.
In this work, we revisit the one- and two-sample testing problems: binary hypothesis testing in which one or both distributions are unknown. For the one-sample test, we provide a more streamlined proof of the asymptotic optimality of Hoeffding's likelihood ratio test, which is equivalent to the threshold test of the relative entropy between the empirical distribution and the nominal distribution. The new proof offers an intuitive interpretation and naturally extends to the two-sample test where we show that a similar form of Hoeffding's test, namely a threshold test of the relative entropy between the two empirical distributions is also asymptotically optimal. A strong converse for the two-sample test is also obtained.