Abstract The projection median as introduced by Durocher and Kirkpatrick (2005); Durocher and Kirkpatrick (2009) is a robust multivariate, nonparametric location estimator. It is a weighted average of points in a sample, where each point’s weight is proportional to the fraction of directions in which that point is a univariate median. The projection median has the highest possible asymptotic breakdown and is easily approximated in any dimension. Previous works have established various geometric properties of the projection median. In this paper we examine further robustness and asymptotic properties of the projection median. We derive the influence function of the projection median which leads to bounds on the maximum bias and contamination sensitivity, as well as an exact expression for the gross error sensitivity. We discuss the degree to which the projection median satisfies these properties relative to other popular robust estimators: specifically, the Zuo projection median and the half-space median. A method for computing the robustness quantities for any distribution and dimension is provided. We then show that the projection median is strongly consistent and asymptotically normal. A method for estimating and computing the asymptotic covariance of the projection median is provided. Lastly, we introduce a large sample multivariate test of location, demonstrating the use of the aforementioned properties. We conclude that the projection median performs very well in terms of the aforementioned robustness quantities but this comes at the cost of dependence on the coordinate system as the projection median is not affine equivariant.

中文翻译：

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投影中位数的稳健性和渐近性

摘要 Durocher 和 Kirkpatrick (2005) 引入的投影中位数；Durocher 和 Kirkpatrick (2009) 是一个稳健的多变量非参数位置估计器。它是样本中点的加权平均值，其中每个点的权重与该点是单变量中位数的方向的分数成正比。投影中位数具有最高可能的渐近分解，并且可以轻松地在任何维度上近似。以前的工作已经建立了投影中位数的各种几何特性。在本文中，我们进一步研究了投影中值的稳健性和渐近特性。我们推导出了投影中值的影响函数，它导致了最大偏差和污染敏感性的界限，以及总误差敏感性的精确表达。我们讨论了投影中位数相对于其他流行的稳健估计器满足这些属性的程度：特别是左投影中位数和半空间中位数。提供了一种用于计算任何分布和维度的稳健性量的方法。然后我们证明投影中位数是强一致的并且渐近正态。提供了一种用于估计和计算投影中值的渐近协方差的方法。最后，我们介绍了一个大样本多变量位置测试，演示了上述属性的使用。我们得出的结论是，投影中值在上述稳健性方面表现非常好，但这是以依赖坐标系为代价的，因为投影中值不是仿射等变的。