乘子U-过程的锐界及其应用
Multiplier U-processes: sharp bounds and applications
摘要：
乘数经验过程理论由于其对各种统计问题的广泛适用性，一直是经典经验过程理论发展的中心议题之一。在本文中，我们开发的理论和工具，研究乘数$U$ -过程，一个自然的高阶推广的乘数经验过程。为此，我们开发了一个乘数不等式，量化的乘数$U$ -过程中的（解耦）对称化$U$-过程的连续性模。新的不等式发现了各种各样的应用，包括（i）乘数和引导中心极限定理的$U$ -过程，（ii）一般理论的引导$M$ -估计的基础上$U$ -统计，和（iii）理论的$M$ -估计一般复杂的抽样设计，再次基于$U$ -统计。
英文摘要：
The theory for multiplier empirical processes has been one of the central topics in the development of the classical theory of empirical processes, due to its wide applicability to various statistical problems. In this paper, we develop theory and tools for studying multiplier $U$-processes, a natural higher-order generalization of the multiplier empirical processes. To this end, we develop a multiplier inequality that quantifies the moduli of continuity of the multiplier $U$-process in terms of that of the (decoupled) symmetrized $U$-process. The new inequality finds a variety of applications including (i) multiplier and bootstrap central limit theorems for $U$-processes, (ii) general theory for bootstrap $M$-estimators based on $U$-statistics, and (iii) theory for $M$-estimation under general complex sampling designs, again based on $U$-statistics.
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