Size and power of pretest procedures

A pre-test 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 locally most powerful tests an attractive expression for the size and power of the pre-test procedure is derived using second order asymptotics. 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.