Robustness properties of S-estimators of multivariate location and shape in high dimension
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高维多元位置与形状 S 估计量的稳健性
- 研究问题
- 为何高维 S 估计量即使崩溃点接近一半，仍可能无法拒绝远离数据主体的离群点，如何改善这种稳健性不足？
- 方法与思路
- 作者重新定义位置与形状估计量，着重同时考虑给定维数和样本量下的崩溃点、远端离群点拒绝能力与计算表现。
- 材料陈述的贡献
- 摘要称新估计量可达到最大崩溃点并拒绝远端离群点，但最大崩溃点需较多计算，改进也伴随效率及敏感性代价。
- 阅读关注点
- 应核对新定义对维数的依赖、污染下算法收敛的适用范围，以及最大崩溃点、离群点拒绝与各敏感性指标的权衡。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要；较少计算获得良好解的陈述不覆盖接近一半的污染，不能与最大崩溃点保证混同。
原文摘要
For the problem of robust estimation of multivariate location and shape, defining S-estimators using scale transformations of a fixed $\rho$ function regardless of the dimension, as is usually done, leads to a perverse outcome: estimators in high dimension can have a breakdown point approaching 50%, but still fail to reject as outliers points that are large distances from the main mass of points. This leads to a form of nonrobustness that has important practical consequences. In this paper, estimators are defined that improve on known S-estimators in having all of the following properties: (1) maximal breakdown for the given sample size and dimension; (2) ability completely to reject as outliers points that are far from the main mass of points; (3) convergence to good solutions with a modest amount of computation from a nonrobust starting point for large (though not near 50%) contamination. However, to attain maximal breakdown, these estimates, like other known maximal breakdown estimators, require large amounts of computational effort. This greater ability of the new estimators to reject outliers comes at a modest cost in efficiency and gross error sensitivity and at a greater, but finite, cost in local shift sensitivity.