17/05/2001 Large Deviations of U- and V- statistics with statistical application

["Giacomo Aletti" <giacomo.aletti@unimi.it>]

Università degli Studi di Milano
Dipartimento di Matematica, 
Via Saldini, 50. 

17 Maggio 2001, ore 15:30, Sala Rappresentanza

LARGE DEVIATIONS OF U- AND V-STATISTICS
WITH STATISTICAL APPLICATIONS

       Yakov Nikitin (University of St.Petersburg)

   The class of U- and V-statistics was introduced in late 40-s by von
Mises and Hoeffding as the generalization of the sample mean. Many
important estimates and test statistics such as the sample variance,
Gini's mean difference, the Wilcoxon, the chi-square and the omega-square
statistics belong to this class.
    Despite many deep results in the limiting theory of U- and V-statistics
the problem of their  large deviation behavior was solved
only for particular kernels. We give the solution for the nondegenerate
and for the "weakly degenerate" case. The kernel is assumed to be bounded,
but no smoothness conditions are imposed. In the nondegenerate case we
correct a wrong theorem of R.Dasgupta (1984), the result in degenerate
case is completely new. We use Sanov's theorem combined with recent
results of Serfling and Wang on large deviation principle for U-statis-
tical empirical measures and solve the corresponding extremal problem
using the theory of implicit analytical operators in an appropriate Banach
space.
   The results could be applied to the calculation of Bahadur, Chernoff
and Hodges-Lehmann efficiencies of numerous nonparametric tests.