The Annals of Statistics

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

Robert P. Sherman

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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

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

First available in Project Euclid: 11 April 2007

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Zentralblatt MATH identifier


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

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


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.

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