U-processes: Rates of convergence. (English) Zbl 0624.60048

Nolan, Deborah; Pollard, David

Ann. Stat. 15, 780-799 (1987).

Let \(\xi_ 1,\xi_ 2,..\). be independent, identically distributed random variables and denote by \[ S_ n(f)=\sum_{1\leq i\neq j\leq n}f(\xi_ i,\xi_ j) \] the U-statistic with respect to the kernel f. The authors obtain almost sure convergence results for \(S_ n(f)\) uniformly over \(f\in F\) where F belongs to certain classes of kernels. Assumptions and proofs are motivated by the corresponding theory for empirical processes, though there are several significant differences in this case. Finally, an application to cross validation in density estimation is given.

Reviewer: M.Denker

MSC:

60F15	Strong limit theorems
62G05	Nonparametric estimation
62E20	Asymptotic distribution theory in statistics

Keywords:

uniform almost sure convergence; U-statistic; empirical processes; cross validation; density estimation