Depth functions and mutidimensional medians on minimal spanning trees

Based on the minimal spanning tree (MST) of the observed data set, the paper introduces new notions of data depth and medians for multivariate data. The MST of a data set of size n is the MST of the complete weighted undirected graph on n vertices, where the edge weights are the pairwise distances of the data points. We study several properties of the MST-based depth functions. We consider the corresponding multidimensional medians, investigate their robustness and computational complexity. An example illustrates the use of the MST-based depth functions.