On Efficiency of First and Second Order

It has been noted by a number of authors that if two tests are asymptotically efficient for the same testing problem, then typically their powers will not only agree to first but also to second order. A general result of this type was given by Pfanzagl (1979) in a paper entitled ’First order efficiency implies second order efficiency’. Because of their technical nature, however, these contributions give little insight into the nature of this phenomenon. The purpose of the present paper is to provide an intuitive understanding of the phenomenon by proving a simple theorem of this kind under mild assumptions.