The Annals of Statistics

$U$-Processes: Rates of Convergence

Deborah Nolan and David Pollard

Full-text: Open access

Abstract

This paper introduces a new stochastic process, a collection of $U$-statistics indexed by a family of symmetric kernels. Conditions are found for the uniform almost-sure convergence of a sequence of such processes. Rates of convergence are obtained. An application to cross-validation in density estimation is given. The proofs adapt methods from the theory of empirical processes.

Article information

Source
Ann. Statist. Volume 15, Number 2 (1987), 780-799.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1176350374

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aos/1176350374

Mathematical Reviews number (MathSciNet)
MR888439

Zentralblatt MATH identifier
0624.60048

Subjects
Primary: 60F15: Strong theorems
Secondary: 62G99: None of the above, but in this section 60G20: Generalized stochastic processes

Keywords
$U$-statistics empirical processes rates of convergence cross-validation reversed submartingale maximal inequality kernel density estimation

Citation

Nolan, Deborah; Pollard, David. $U$-Processes: Rates of Convergence. The Annals of Statistics 15 (1987), no. 2, 780--799. doi:10.1214/aos/1176350374. http://projecteuclid.org/euclid.aos/1176350374.


Export citation