The penalty in data driven Neyman's tests

Data driven Neyman's tests are based on two elements: Neyman's smooth tests in finite dimensional submodels and a selection rule to choose the "right'' submodel. As selection rule usually (a modification of) Schwarz's rule is applied. In this paper we consider data driven Neyman's tests with selection rules allowing also other penalties than the one in Schwarz's rule. It is shown that the nice properties of consistency against very large classes of alternatives and the more deep result of asymptotic optimality in the sense of vanishing shortcoming continue to hold for other penalties as well, including the one corresponding to Akaike's selection rule.