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

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.