The lens depth of a point has been recently extended to general metric spaces, which is not the case for most depths. It is defined as the probability of being included in the intersection of two random balls centred at two random points X and Y, with the same radius d(X, Y). We study the consistency in Hausdorff and measure distance, of the level sets of the empirical lens depth, based on an iid sample on a general metric space. We also prove that the boundary of the empirical level sets are consistent estimators of their population counterparts, and analyze two real-life examples
翻译:最近,一个点的透镜深度已扩大到一般计量空间,而大多数深度的情况并非如此。它的定义是,在两个随机点X和Y的两个随机点以X和Y为中心,以同一半径d(X,Y)为同一点的两个随机球的交叉点中,我们根据一个一般计量空间的iid样本,在Hausdorf和测量距离方面研究实验透镜深度水平组的一致性和测量距离。我们还证明,经验级组的边界是其人口对应方的一致估测者,我们分析了两个真实生活的例子。