Third order efficiency of conditional tests for exponential families

Let P[eta], [eta] = ([theta], [gamma]) [set membership, variant] [Theta] - [Gamma] [subset of] - k, be a (k + 1)-dimensional exponential family. Let [phi]n*, n [set membership, variant] , be an optimal similar test for the hypothesis {P([theta],[gamma])n: [gamma] [set membership, variant] [Gamma]} ([theta] [set membership, variant] [Theta] fixed) against alternatives P([theta]1,[gamma]1)n, [theta]1 > [theta], [gamma]1 [set membership, variant] [Gamma]. It is shown that ([phi]n*)n[set membership, variant] is third order efficient in the class of all test-sequences that are asymptotically similar of level [alpha] + o(n-1) (locally uniformly in the nuisance parameter [gamma]).