The aim of this volume is to present an introduction to the modern asymptotic theory of testing statistical hypotheses and to give an overview of methods used for the investigation of the higher order asymptotics for the widely used statistics such as linear combinations of order statistics (L-statistics), linear rank statistics ( R-statistics), and U-statistics, etc. Only purely theoretical, mathematical aspects of the subjects are considered. In the book, the problem of testing a simple hypothesis concerning a one-dimensional parameter against a sequence of close one-sided alternatives is considered, based on the so called Pitman's approach where both size and power of tests are separated from zero and one, respectively, for second order deficiency, asymptotic deficiency and power loss. The book provides base statements, which are provided with the use of a technically rather simple, though more clear and visible method. Main attention is focused on tests based on linear rank statistics, linear combinations of order statistics and U-statistics. This volume should be of value and interest to specialists in probability theory and mathematical statistics.

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