Computational Statistics

, Volume 32, Issue 4, pp 1395–1410 | Cite as

Fast implementation of the Tukey depth

Original Paper


Tukey depth function is one of the most famous multivariate tools serving robust purposes. It is also very well known for its computability problems in dimensions \(p \ge 3\). In this paper, we address this computing issue by presenting two combinatorial algorithms. The first is naive and calculates the Tukey depth of a single point with complexity \(O\left( n^{p-1}\log (n)\right) \), while the second further utilizes the quasiconcave of the Tukey depth function and hence is more efficient than the first. Both require very minimal memory and run much faster than the existing ones. All experiments indicate that they compute the exact Tukey depth.


Tukey depth Quasiconcave Combinatorial property Fast computation 



The author thanks Prof. Mosler, K. and Dr. Mozharovskyi, P. for their valuable discussions during the preparation of this manuscript. The author also greatly appreciates two anonymous reviewers for their careful reading and insightful comments, which led to many improvements in this paper. This research is supported by NSFC of China (No. 11601197, 11461029, 71463020), NSF of Jiangxi Province (No. 20161BAB201024, 20151BAB211016), and the Key Science Fund Project of Jiangxi provincial education department (No. GJJ150439).

Compliance with ethical standards

Conflict of interest

I am the sole author of this manuscript. This research involves no human participants and/or animals, and has no conflict of interest.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of StatisticsJiangxi University of Finance and EconomicsNanchangChina
  2. 2.Research Center of Applied StatisticsJiangxi University of Finance and EconomicsNanchangChina

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