基于投影深度的多维裁剪
Multidimensional trimming based on projection depth
摘要：
作为位置参数的估计，一元修剪均值以其稳健性和有效性而闻名。它们可以作为样本均值的稳健替代品，同时在正态和重尾模型中具有高效率。提出了一种基于投影深度诱导区域的多维裁剪均值算法。从影响函数和有限样本崩溃点两个方面研究了这些深度裁剪方法的鲁棒性。影响函数反映了估计量的局部鲁棒性，而崩溃点则反映了估计量的全局鲁棒性。它被发现，投影深度修剪的手段是非常强大的本地以及全球。通过深度诱导区域的方向半径的渐近性的深度修剪的平均值进行了研究。得到了深度截尾均值的强相合性、渐近表示和极限分布。相对于平均值和其他主要竞争者，深度修剪平均值在正态或对称模型中非常有效，并且在这些模型被污染时非常有效。仿真研究证实了有限样本下渐近效率结果的有效性。
英文摘要：
As estimators of location parameters, univariate trimmed means are well known for their robustness and efficiency. They can serve as robust alternatives to the sample mean while possessing high efficiencies at normal as well as heavy-tailed models. This paper introduces multidimensional trimmed means based on projection depth induced regions. Robustness of these depth trimmed means is investigated in terms of the influence function and finite sample breakdown point. The influence function captures the local robustness whereas the breakdown point measures the global robustness of estimators. It is found that the projection depth trimmed means are highly robust locally as well as globally. Asymptotics of the depth trimmed means are investigated via those of the directional radius of the depth induced regions. The strong consistency, asymptotic representation and limiting distribution of the depth trimmed means are obtained. Relative to the mean and other leading competitors, the depth trimmed means are highly efficient at normal or symmetric models and overwhelmingly more efficient when these models are contaminated. Simulation studies confirm the validity of the asymptotic efficiency results at finite samples.
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