Statistical Papers

, Volume 57, Issue 1, pp 205–234 | Cite as

Robust estimation of location and concentration parameters for the von Mises–Fisher distribution

  • Shogo KatoEmail author
  • Shinto Eguchi
Regular Article


Robust estimation of location and concentration parameters for the von Mises–Fisher distribution is discussed. A key reparametrisation is achieved by expressing the two parameters as one vector on the Euclidean space. With this representation, we first show that maximum likelihood estimator for the von Mises–Fisher distribution is not robust in some situations. Then we propose two families of robust estimators which can be derived as minimisers of two density power divergences. The presented families enable us to estimate both location and concentration parameters simultaneously. Some properties of the estimators are explored. Simple iterative algorithms are suggested to find the estimates numerically. It is shown that the presented approaches can be utilised to estimate either the location or concentration parameter. A comparison with the existing robust estimators is given as well as discussion on difference and similarity between the two proposed estimators. A simulation study is made to evaluate finite sample performance of the estimators. We apply the proposed methods to a sea star dataset and discuss the selection of the tuning parameters and outlier detection.


Directional data Divergence Influence function  Outlier detection Robust inference 

Mathematics Subject Classification

62H11 62F35 62E15 



The authors are grateful to the Editor and two referees for helpful comments which improve the quality of the paper. The work of the first author was supported by JSPS KAKENHI Grant Number 25400218. That of the second author was supported by Japan Science and Technology Agency (JST), Core Research for Evolutionary Science and Technology (CREST).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.The Institute of Statistical MathematicsTachikawaJapan

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