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Maximal inequalities for degenerate $$U$$-processes with applications to optimization estimators. (English) Zbl 0798.60021
Maximal inequalities for degenerate $$U$$-processes of order $$k$$, $$k \geq 1$$, are established. The results rest on a moment inequality for $$k$$th- order forms and the 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.

##### MSC:
 60E15 Inequalities; stochastic orderings 60G20 Generalized stochastic processes 62E20 Asymptotic distribution theory in statistics
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