Some important problems of probability and statistics can be reduced to the evaluation of supremum of some homogeneous functional defined on the Strasses ball in the space of smooth functions on the square. We give the solution of this extremal problem in two particular cases: when the...

The asymptotics of large deviation probabilities for two-sample non degenerate von Mises functionals and U- statistics of arbitrary degree is investigated. We find explicitly the main term of this asymptotics. Some examples motivated by nonparametric statistics are given. The result may be...

Many well-known goodness-of-fit and symmetry tests are based on U- et V-statistics. Recent progress in the analysis of their large deviations enables to find new expressions for their local Bahadur efficiency in case of bounded kernels and to formulate the conditions of local optimality. For...