On the Asymptotic Efficiency of Tests in Hypothesis Testing Problems

The paper contains an introduction to the asymptotic theory of hypothesis testing and a review of recent results of the author and his students. We consider only the asymptotic approach, for which with increasing sample size n the test size is separated from zero, and the sequences of local alternatives, for which the power is separated from one. This paper focuses on asymptotically efficient tests when testing a simple hypothesis in the case of a one-parameter family. We study the difference between the powers of the best and asymptotically efficient tests. This difference is closely related to the notion of asymptotic test deficiency. We consider the formula for the limiting deviation of the power of asymptotically optimal test from the power of the best test in the case of Laplace distribution. Due to the irregularity of the Laplace distribution, this deviation is of order n −1/2, in contrast to the usual regular families for which this order is n −1. We also study the Bayesian settings and the case of increasing parameter dimension.