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

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

SEMINARIO DI MATEMATICA APPLICATA
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Giovedì 17 Maggio 2001, ore 15.30 precise

Aula Rappresentanza, Dipartimento di Matematica (Via Saldini 50, Milano)
LARGE DEVIATIONS OF U- AND V-STATISTICS WITH STATISTICAL APPLICATIONS
Yakov Nikitin (Unuversity of St.Petersburg)

The class of U- and V -statistics was introduced in late 40-s by von Mises and Ho-effding
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 re-sults
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 nonde-generate
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 Ser ing and Wang on large
deviation principle for U-statistical 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.

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La home page del seminario `e http://www.mat.unimi.it/ ¢ pavarino/sma/
Il seminario `e organizzato dal Dipartimento di Matematica dell’Universit` a di Milano
in collaborazione con MIRIAM (MIlan Research group in Industrial and Applied
Mathematics) e il Dottorato di Ricerca MACRO (MAtematica Computazionale e
Ricerca Operativa). Per ulteriori informazioni contattare Luca Pavarino o Kevin
Payne.