Weak Convergence Analysis of Asymptotically Optimal Hypothesis Tests

In recent years, solutions to various hypothesis testing problems in the asymptotic setting have been proposed using the results from large deviation theory. Such tests are optimal in terms of appropriately defined error exponents. For the practitioner, however, error probabilities in the finite sample size setting are more important. In this paper, we show how results on weak convergence of the test statistic can be used to obtain better approximations for the error probabilities in the finite sample size setting. While this technique is popular among statisticians for common tests, we demonstrate its applicability for several recently proposed asymptotically optimal tests, including tests for robust goodness of fit, homogeneity tests, outlier hypothesis testing, and graphical model estimation.