Approximations for multivariate U-statistics

On normal approximations to U-statistics

Let X1, . . . ,Xn be i.i.d. random observations. Let S = L + T be a U -statistic of order k ≥ 2, where L is a linear statistic having asymptotic normal distribution, and T is a stochastically smaller statistic. We show that the rate of convergence to normality for S can be simply expressed as the rate of convergence to normality for the linear part L plus a correction term, (varT) ln(varT), und...

Generalized Multivariate Rank Type Test Statistics via Spatial U-Quantiles

The classical univariate sign and signed rank tests for location have been extended over the years to the multivariate setting, including recent robust rotation invariant “spatial” versions. Here we introduce a broad class of rotation invariant multivariate spatial generalized rank type test statistics. For a given inference problem not restricted to location, the test statistics are linked thr...

Multivariate Dispersive Ordering of Generalized Order Statistics

The concept of generalized order statistics (GOSs) was introduced as a unified approach to a variety of models of ordered random variables. The purpose of this paper is to investigate condi- tions on the underlying distribution functions and the parameters on which GOSs are based, to establish Shaked-Shanthikumar multivari- ate dispersive ordering of GOSs from one sample and Khaledi-Kochar mu...

Decoupling Inequalities for the Tail Probabilities of Multivariate U-statistics by

In this paper we present a decoupling inequality that shows that multivariate Ustatistics can be studied as sums of (conditionally) independent random variables. This result has important implications in several areas of probability and statistics including the study random graphs and multiple stochastic integration. More precisely, we get the following result: Theorem 1. Let {Xj} be a sequence...

Robust Statistics for Multivariate Methods