A pretest procedure consists of a preliminary test on a nuisance parameter, investigating whether it equals a given value or not, followed by the main testing problem on the parameter of interest. In case of acceptance of the preliminary test, the main test is applied in the restricted family with the given value of the nuisance parameter, while otherwise the test is performed in the complete family, including the nuisance parameter. For an appropriate class of tests, containing all standard first-order optimal tests, an attractive expression for the difference in size and power between the pretest procedure and the test in the complete family is derived using second-order asymptotics. For a very great part the result is the same for all members of the class. From this expression considerable insight can be obtained in a qualitative and quantitative sense. The results can be applied easily as is illustrated by a number of practical examples, where also the accuracy of the approximations is seen from comparison with numerical results.
"Size and power of pretest procedures." Ann. Statist. 28 (1) 195 - 214, February 2000. https://doi.org/10.1214/aos/1016120369