Asymptotic Power Functions

Consider the problem of testing a null hypothesis versus the alternative The power function of a test that rejects the null hypothesis if a test statistic falls into a critical region Kn is the function which gives the probability of rejecting the null hypothesis. The test is of level if its size does not exceed A sequence of tests is called asymptotically of level a if

(An alternative definition is to drop the supremum and require only thatfor everyA test with power function is better than a test with power function

The aim of this chapter is to compare tests asymptotically. We consider sequences of tests with power functions and and wish to decide which of the sequences is best as . Typically, the tests corresponding to a sequence are of the same type. For instance, they are all based on a certain U -statistic or rank statistic, and only the number of observations changes with n. Otherwise the comparison would have little relevance. A first idea is to consider limiting power functions of the form

If this limit exists for all and the same is true for the competing tests then the sequence is better than the sequenceif the limiting power functionis better than the limiting power function. It turns out that this approach is too naive. The limiting power functions typically exist, but they are trivial and identical for all reasonable sequences of tests.