Power gain by pre-testing?

The aim of this paper is to study whether it is possible to gain power by pre-testing, and to give insight in when this occurs, to what extent, and, at which price. A pre-test procedure consists of a preliminary test which tests a particular property of a given restricted model, followed by a main test for the main hypothesis regarding the parameter of interest. After acceptance by the preliminary test, a basic main test is used in the restricted model. After rejection by the preliminary test, a more general main test is used which does not use prior information about the underlying distribution. The procedure is analyzed in the model against which the preliminary test protects. For classes of tests including the standard first-order optimal tests, a transparent expression is given for the power and size difference of the pre-test procedure compared to the power and (correct) size of the second main test. This expression is based on second-order asymptotics and gives qualitative and quantitative insight in the behaviour of the procedure. It shows that substantial power gain, not merely due to size violation, is possible if the second main test really differs from the basic main test. The smaller the correlation between the two main tests, the larger the power gain.