an:04013653
Zbl 0624.60048
Nolan, Deborah; Pollard, David
U-processes: Rates of convergence
EN
Ann. Stat. 15, 780-799 (1987).
00152866
1987
j
60F15 62G05 62E20
uniform almost sure convergence; U-statistic; empirical processes; cross validation; density estimation
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.
M.Denker