Most mean powerful test of a composite null against a composite alternative

This paper presents a new approach to testing a composite null against a composite alternative hypothesis. The test is based on the generalized Neyman-Pearson lemma and maximizes average power subject to controlling average size over different subsets of the null hypothesis parameter space. Our focus is on testing problems that can be reduced through invariance or other arguments to composite hypotheses involving one parameter. We illustrate the new test procedure by applying it to the problem of testing for MA(1) disturbances against AR(1) disturbances in the linear regression model, with encouraging results. The standard approach of controlling the maximum size is typically difficult and time consuming. Controlling average size over subregions selected to reduce variability in size does seem to be an alternative worthy of investigation.