Higher Order Asymptotic Optimality in Testing Problems With Nuisance Parameters

We consider testing hypotheses about the location parameter of a symmetric distribution when a finite-dimensional nuisance parameter is present. For local alternatives, we study the power loss of asymptotically efficient tests in this problem, which is the difference between the power of the most powerful test for a given value of the nuisance parameter (as if it were known) and the power of the test at hand. The power loss is typically of order n \Gamma1 and is closely related to the deficiency of the test. In particular, we obtain the lower bound for the power loss in a locally asymptotically minimax sense similar to that used in the estimation theory and indicate a test on which this bound is attained. This bound corresponds to the envelope power function obtained by Pfanzagl and Wefelmeyer (1978) for test statistics of a specific structure