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In this paper, a brief survey of the asymptotic theory of hypotheses testing is presented and some of the author’s recent results are given. The survey is not intended to be complete; it contains mainly results related to the author's interests (for detailed proofs, see [51]). A detailed review of this field can be found in Pfanzagl [45], Pfanzagl and Wefelmeyer [46, 47], Chibisov [23], and Götze and Milbrodt [25].

We consider the asymptotic approach with the probabilities of errors of first and second kind being bounded away from zero and therefore we study the power of tests against local alternatives. Special attention is paid to asymptotically efficient tests for testing a simple hypothesis concerning a univariate parameter. We shall consider only “regular” families for which local alternatives approach the hypothesis at a rate of 

hypotheses testing, asymptotic expansions, deficiency, efficiency, power loss, test, level, errors of first and second kind.