A note on asymptotic expansions for the power of perturbed tests

For simplicity or tractability reasons one sometimes uses modified test statistics, which differ from the original ones up to O p ( a n ) terms with a n → 0 . In this note, some technical conditions are provided under which a corresponding expansion for the powers of such perturbed tests holds. The necessity of some of these conditions is discussed and illustrated by examples. An application to invariant testing multivariate normality is presented.