The Annals of Statistics

Maximal Inequalities for Degenerate $U$-Processes with Applications to Optimization Estimators

Robert P. Sherman

Full-text: Open access

Abstract

Maximal inequalities for degenerate $U$-processes of order $k, k \geq 1$, are established. The results rest on a moment inequality (due to Bonami) for $k$th-order forms and on extensions of chaining and symmetrization inequalities from the theory of empirical processes. Rates of uniform convergence are obtained. The maximal inequalities can be used to determine the limiting distribution of estimators that optimize criterion functions having $U$-process structure. As an application, a semiparametric regression estimator that maximizes a $U$-process of order 3 is shown to be $\sqrt n$-consistent and asymptotically normally distributed.

Article information

Source
Ann. Statist., Volume 22, Number 1 (1994), 439-459.

Dates
First available in Project Euclid: 11 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176325377

Digital Object Identifier
doi:10.1214/aos/1176325377

Mathematical Reviews number (MathSciNet)
MR1272092

Zentralblatt MATH identifier
0798.60021

JSTOR
links.jstor.org

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 60E15: Inequalities; stochastic orderings 60G20: Generalized stochastic processes 60G99: None of the above, but in this section

Keywords
Maximal inequality degenerate $U$-processes empirical processes chaining symmetrization polynomial classes of sets Euclidean classes of functions optimazation estimator semiparametric estimation generalized regression model

Citation

Sherman, Robert P. Maximal Inequalities for Degenerate $U$-Processes with Applications to Optimization Estimators. Ann. Statist. 22 (1994), no. 1, 439--459. doi:10.1214/aos/1176325377. https://projecteuclid.org/euclid.aos/1176325377


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