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

Let ξ 1 ,ξ 2 ,··. be independent, identically distributed random variables and denote by

S n (f)= 1ijn f(ξ i ,ξ j )

the U-statistic with respect to the kernel f. The authors obtain almost sure convergence results for S n (f) uniformly over fF 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:
60F15Strong limit theorems
62G05Nonparametric estimation
62E20Asymptotic distribution theory in statistics