A New Method for Testing Separate Families of Hypotheses

Theoretical and empirical evidence are reported showing that the tests given in Cox (1961) and Williams (1970a, b) for the problem of testing separate families of hypotheses suffer from fundamental difficulties. Neither test is guaranteed to be asymptotically level a, even for very smooth problems. A new method proposed here as an alternative works adaptively by repeated application of the parametric bootstrap idea over slowly shrinking confidence regions. Theoretical properties of the method are derived, and its finite-sample behavior is studied in a number of examples.