Decoupling of U-Statistics and U-Processes

Abstract

U-statistics, first considered by Halmos (1946) in connection with unbiased statistics, and formally introduced by Hoeffding (1948), are defined as follows: Given an i.i.d. sequence of random variables {X _i } ^∞_i=1 with values in a measurable space (S,S), and a measurable function h : S^m → ℝ, the U-statistics of order m and kernel h based on the sequence {X _i}are

$$ {{U}_{n}}\left( h \right)\frac{{\left( {n - m} \right)!}}{{n!}}\sum\limits_{{\left( {{{i}_{{l,...,}}}{{i}_{m}}} \right) \in I_{n}^{m}}} {h\left( {{{X}_{{i,....,}}}{{X}_{{{{i}_{m}}}}}} \right)} , n \geqslant m, $$

(3.0.1)

where

$$ I_n^m = \left\{ {(i_1 , \ldots ,i_m ):i_j \in \mathbb{N},1 \leqslant i_j \leqslant n,i_j \ne i_k if j \ne k} \right\}. $$

These objects appear often in statistics either as unbiased estimators of parameters of interest or, perhaps more often, as components of higher-order terms in expansions of smooth statistics (von Mises expansion, delta-method). Particularly in connection with von Mises expansions, it is sometimes convenient to also consider U-processes indexed by families 3E of kernels, that is, collections of U-statistics{U _n(h): h ∈ H}.Decoupling inequalities have had a major role in recent advances on the asymptotic theory of U-statistics and U-processes. In this chapter we present the decoupling theory and Chapters 4 and 5 are devoted to its applications to limit theorems.