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Robust M-Estimators on Spheres

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  • Ko, D. J.
  • Chang, T.

Abstract

We introduce M-estimators for location and concentration parameters of von Mises-Fisher distributions on unit spheres. These include the directional mean, normalized spatial median, spherical median, and the mle of the concentration parameter. We find the influence functions and asymptotic distributions of such estimators and give necessary and sufficient conditions under which the M-estimators become SB-robust. SB-robust M-estimators, which are optimal in a sense similar to Hampel, are proposed. We discuss both simultaneous estimation of location and concentration and estimation of one parameter when the other is known. The behavior of the optimal estimators, together with several alternatives, under extreme contamination and moderate sample sizes is studied using simulation. An example, previously studied by Fisher, Lewis, and Embleton (1987, Analysis of Spherical Data, Cambridge Univ. Press, Cambridge, UK), dealing with remanent magnetization is reanalyzed using these techniques.

Suggested Citation

  • Ko, D. J. & Chang, T., 1993. "Robust M-Estimators on Spheres," Journal of Multivariate Analysis, Elsevier, vol. 45(1), pages 104-136, April.
    • Handle: RePEc:eee:jmvana:v:45:y:1993:i:1:p:104-136
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    Cited by:

    1. Shogo Kato & Shinto Eguchi, 2016. "Robust estimation of location and concentration parameters for the von Mises–Fisher distribution," Statistical Papers, Springer, vol. 57(1), pages 205-234, March.
    2. Luca Greco & Giovanni Saraceno & Claudio Agostinelli, 2021. "Robust Fitting of a Wrapped Normal Model to Multivariate Circular Data and Outlier Detection," Stats, MDPI, vol. 4(2), pages 1-18, June.
    3. Giovanni Saraceno & Claudio Agostinelli & Luca Greco, 2021. "Robust estimation for multivariate wrapped models," METRON, Springer;Sapienza Università di Roma, vol. 79(2), pages 225-240, August.
    4. Dabo-Niang, Sophie & Thiam, Baba & Verdebout, Thomas, 2022. "Asymptotic efficiency of some nonparametric tests for location on hyperspheres," Statistics & Probability Letters, Elsevier, vol. 188(C).
    5. Arnab Kumar Laha & A. C. Pravida Raja & K. C. Mahesh, 2019. "SB-robust estimation of mean direction for some new circular distributions," Statistical Papers, Springer, vol. 60(3), pages 877-902, June.
    6. Chang, Ted & Tsai, Ming-Tien, 2003. "Asymptotic relative Pitman efficiency in group models," Journal of Multivariate Analysis, Elsevier, vol. 85(2), pages 395-415, May.
    7. Kirschstein, Thomas & Liebscher, Steffen & Pandolfo, Giuseppe & Porzio, Giovanni C. & Ragozini, Giancarlo, 2019. "On finite-sample robustness of directional location estimators," Computational Statistics & Data Analysis, Elsevier, vol. 133(C), pages 53-75.

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