Influence function and maximum bias of projection depth based estimators
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投影深度估计量的影响函数与最大偏差
- 研究问题
- 除了有限样本崩溃点，如何通过影响函数、最大偏差及污染敏感度，全面刻画投影中位数和投影深度加权均值的稳健性？
- 方法与思路
- 作者推导并研究两类位置估计量的影响函数、精确最大偏差、污染及总误差敏感度，同时建立相关量的尖锐上界。
- 材料陈述的贡献
- 摘要称，相关比较显示这些估计量兼具局部与全局稳健性，具有较强竞争力；加权均值类别包含 Stahel–Donoho 估计量。
- 阅读关注点
- 核对投影深度和权重的定义、基准分布与污染模型、精确表达及上界的条件，以及稳健性比较使用的竞争方法和指标。
- 阅读边界
- 来源为稿件版本未注明的 OpenAlex 索引摘要；具体分布、污染条件及比较结果未提供，不能将稳健性竞争力解释为统一效率或风险优势。
原文摘要
Location estimators induced from depth functions increasingly have been pursued and studied in the literature. Among them are those induced from projection depth functions. These projection depth based estimators have favorable properties among their competitors. In particular, they possess the best possible finite sample breakdown point robustness. However, robustness of estimators cannot be revealed by the finite sample breakdown point alone. The influence function, gross error sensitivity, maximum bias and contamination sensitivity are also important aspects of robustness. In this article, we study these other robustness aspects of two types of projection depth based estimators: projection medians and projection depth weighted means. The latter includes the Stahel-Donoho estimator as a special case. Exact maximum bias, the influence function, and contamination and gross error sensitivity are derived and studied for both types of estimators. Sharp upper bounds for the maximum bias and the influence functions are established. Comparisons based on these robustness criteria reveal that the projection depth based estimators enjoy desirable local as well as global robustness and are very competitive among their competitors.