Projection-based depth functions and associated medians
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基于投影的深度函数及其诱导中位数
- 研究问题
- 如何构造具有良好结构与渐近性质的投影深度函数及多元中位数，并兼顾有限样本稳健性和效率？
- 方法与思路
- 作者研究投影深度、深度区域与轮廓及其诱导的仿射等变中位数，建立样本收敛结果，并与半空间中位数比较。
- 材料陈述的贡献
- 摘要称，论文建立一致性与极限分布等结果；适当选择一元位置和尺度估计量时，投影中位数具有较高崩溃点与效率。
- 阅读关注点
- 应核对投影深度的定义、一元估计量选择、均匀一致性条件，以及崩溃点和相对效率比较所采用的设定。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明；优势依赖适当的一元估计量选择，摘要未说明全部条件或比较数值。
原文摘要
A class of projection-based depth functions is introduced and studied. These projection-based depth functions possess desirable properties of statistical depth functions and their sample versions possess strong and order $\sqrt{n}$ uniform consistency. Depth regions and contours induced from projection-based depth functions are investigated. Structural properties of depth regions and contours and general continuity and convergence results of sample depth regions are obtained. Affine equivariant multivariate medians induced from projection-based depth functions are probed. The limiting distributions as well as the strong and order $\sqrt{n}$ consistency of the sample projection medians are established. The finite sample performance of projection medians is compared with that of a leading depth-induced median, the Tukey halfspace median (induced from the Tukey halfspace depth function). It turns out that, with appropriate choices of univariate location and scale estimators, the projection medians have a very high finite sample breakdown point and relative efficiency, much higher than those of the halfspace median. Based on the results obtained, it is found that projection depth functions and projection medians behave very well overall compared with their competitors and consequently are good alternatives to statistical depth functions and affine equivariant multivariate location estimators, respectively.