How powerful are data driven score tests for uniformity
We construct a new class of data driven tests for uniformity, which have greater average power than existing ones for finite samples. Using a simulation study, we show that these tests as well as some “optimal maximum test” attain an average power close to the optimal Bayes test. Finally, we prove that, in the middle range of the power function, the loss in average power of the “optimal maximum test” with respect to the Neyman–Pearson tests, constructed separately for each alternative, in the Gaussian shift problem can be measured by the Shannon entropy of a prior distribution. This explains similar behaviour of the average power of our data driven tests.