Regression Depth and Center Points

Abstract.

We show that, for any set of n points in d dimensions, there exists a hyperplane with regression depth at least $\lceil n/(d+1)\rceil$ , as had been conjectured by Rousseeuw and Hubert. Dually, for any arrangement of n hyperplanes in d dimensions there exists a point that cannot escape to infinity without crossing at least $\lceil n/(d+1)\rceil$ hyperplanes. We also apply our approach to related questions on the existence of partitions of the data into subsets such that a common plane has nonzero regression depth in each subset, and to the computational complexity of regression depth problems.