增长阶U过程的偏差不等式及其在半空间深度中的应用
Deviation inequalities for U-processes of growing order, with applications to half-space depth
AI总结：
该研究针对阶数随样本量增长的无穷阶U过程推导偏差不等式，将其应用于多元深度，建立了基于Tukey中位数的多元Hodges-Lehmann估计量的亚高斯集中不等式并发现其渐近分布的有趣现象。
AI中文摘要：
我们证明了无穷阶U过程的新偏差不等式，即阶数可随样本量增长的U过程。我们的目标是获得尾界对过程阶数的最尖锐依赖关系，并刻画过程呈现亚高斯偏差的最大范围。我们推导了由欧几里得类（包括VC-子图函数类）索引的过程的闭式不等式。随后我们将结果的统计应用例证到多元深度中。特别地，我们引入了基于Tukey中位数的Hodges-Lehmann估计量的多元类似物，为其与均值的偏差建立了带显式常数的亚高斯集中不等式，并确定了关于其渐近分布的一个有趣现象。
英文摘要：
We prove new deviation inequalities for infinite-order U-processes, that is, U-processes whose order may increase with the sample size. Our goal is to obtain the sharpest possible dependence of the tail bounds on the order of the process, as well as to characterize the largest range in which the process exhibits sub-Gaussian deviations. We derive closed-form inequalities for processes indexed by Euclidean classes, including VC-subgraph classes of functions. We then illustrate the statistical applications of our results to multivariate depth. In particular, we introduce a multivariate analogue of the Hodges-Lehmann estimator based on Tukey's median, establish sub-Gaussian concentration inequalities with explicit constants for its deviations from the mean, and identify an interesting phenomenon concerning its asymptotic distribution.
