Statistical depths provide a fundamental generalization of quantiles and medians to data in higher dimensions. This paper proposes a new type of globally defined statistical depth, based upon control theory and eikonal equations, which measures the smallest amount of probability density that has to be passed through in a path to points outside the support of the distribution: for example spatial infinity. This depth is easy to interpret and compute, expressively captures multi-modal behavior, and extends naturally to data that is non-Euclidean. We prove various properties of this depth, and provide discussion of computational considerations. In particular, we demonstrate that this notion of depth is robust under an aproximate isometrically constrained adversarial model, a property which is not enjoyed by the Tukey depth. Finally we give some illustrative examples in the context of two-dimensional mixture models and MNIST.
Robust Multivariate Estimation Based On Statistical Depth Filters
在经典的污染模型中，如粗差（Huber和Tukey污染模型或Case-wise污染模型），观测值被认为是被识别为离群值或非离群值的单元。当考虑的变量数量相对较少时，该模型非常有用。Alqallaf et al. [2009]显示了这种方法对大量变量的限制，并引入了独立污染模型（Cell-wise Contamination），其中现在细胞是被识别为离群值或非离群值的单位。同时处理这两种类型的污染的一种方法是从数据集中过滤出受污染的单元格，然后应用能够处理逐例离群值和缺失值的鲁棒程序。在这里，我们开发了一个通用框架，以基于统计数据深度函数在任何维度上构建过滤器。我们证明了以前的方法，例如Agostinelli et al. [2015 a]和Leung et al. [2017]，是特殊情况。我们通过使用半空间深度来说明我们的方法。
LTCI, Télécom Paris, Institut Polytechnique de Paris（巴黎电信学院，巴黎理工学院）；Université Paris Cité, Université Paris Saclay, ENS Paris Saclay, CNRS, SSA, INSERM, Centre Borelli（巴黎西岱大学，巴黎萨克雷大学，巴黎高等师范学校巴黎萨克雷分校，法国国家科学研究中心，SSA，INSERM，博尔莱中心）
Bias robustness of depth estimators in multivariate settings
FAMAF, National University of Córdoba（科尔多瓦国立大学 FAMAF）；CIEM and CONICET, Argentina（阿根廷 CIEM 和 CONICET）；Department of Mathematics, National University of Río Cuarto, Argentina（里奥夸尔托国立大学数学系）
