## The Annals of Statistics

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

Robert P. Sherman

#### 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

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
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