A note on the relation between local power and robustness to misspecification

A simple example is provided that shows that one test can strictly dominate another test in both local power and its robustness to asymptotic size distortion under local model misspecification, with both tests having asymptotic size equal to nominal size under correct model specification. Robustness of the asymptotic size to model misspecification is therefore an alternative criterion to choosing among competing tests that can offer additional discriminatory insight over local power.