Asymptotic Theory of Tests and Interval Estimators

Abstract

The present chapter studies the higher-order asymptotic theory of statistical tests and interval (or region) estimators with or without nuisance parameters. The power function of a test is determined by the geometrical features of the boundary of its critical region. It is proved that a first-order efficient test is automatically second-order efficient, but there is in general no third-order uniformly most powerful test. The third-order power loss functions are explicitly given for various widely used first-order efficient tests. The results demonstrate the universal characteristics of these tests, not depending on a specific model M. We also give the characteristics of the conditional test conditioned on the asymptotic ancillary. The third-order characteristics of interval estimators are also shown. For the sake of simplicity, we maily treat a one-dimensional model, and the multi-dimensional generalization is explained shortly.