On Asymptotically Optimal Tests

Sequences of tests with error $\exp(-nA)$ of the first type are investigated. It is shown that the error of the second type of such a sequence of tests is bounded by $\exp(- nB)$ where $B$ is determined by the Kullback-Leibler information distance of the hypotheses tested. The information distance between the empirical measure and the null-hypothesis on a finite partition of the sample space is proposed to use as a test statistic. A sufficient condition is given which ensures that this test has error of the second type about $\exp(- nB)$ with the best possible $B$. The exact Bahadur slope of the proposed statistic is investigated.