Power gain by pre-testing?

The aim of this paper is to study whether it is possible to gain power by pretesting,
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 rst-order optimal tests, a transparent expression is given for the
power and size dierence 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 behavior of the
procedure. It shows that substantial power gain, not merely due to size violation,
is possible if the second main test really diers from the basic main test. The
smaller the correlation between the two main tests, the larger the power gain.