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A unified approach to decision-theoretic properties of the MLEs for the mean directions of several Langevin distributions

Author

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  • Kanika,
  • Kumar, Somesh
  • SenGupta, Ashis

Abstract

The two-parameter Langevin distribution has been widely used for analyzing directional data. We address the problem of estimating the mean direction in its Cartesian and angular forms. The equivariant point estimation is introduced under different transformation groups. The maximum likelihood estimator (MLE) is shown to satisfy many decision theoretic properties such as admissibility, minimaxity, the best equivariance and risk-unbiasedness under various loss functions. Moreover, it is shown to be unique minimax when the concentration parameter is assumed to be known. These results extend and unify earlier results on the optimality of the MLE. These findings are also established for the problem of simultaneous estimation of mean directions of several independent Langevin populations. Further, estimation of a common mean direction of several independent Langevin populations is studied. A simulation study is carried out to analyze numerically the risk function of the MLE.

Suggested Citation

  • Kanika, & Kumar, Somesh & SenGupta, Ashis, 2015. "A unified approach to decision-theoretic properties of the MLEs for the mean directions of several Langevin distributions," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 160-172.
    • Handle: RePEc:eee:jmvana:v:133:y:2015:i:c:p:160-172
    DOI: 10.1016/j.jmva.2014.09.002
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    References listed on IDEAS

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    1. Hendriks, Harrie & Landsman, Zinoviy & Ruymgaart, Frits, 1996. "Asymptotic Behavior of Sample Mean Direction for Spheres," Journal of Multivariate Analysis, Elsevier, vol. 59(2), pages 141-152, November.
    2. Ashis Sengupta & Ranjan Maitra, 1998. "On Best Equivariance and Admissibility of Simultaneous MLE for Mean Direction Vectors of Several Langevin Distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 50(4), pages 715-727, December.
    3. Perron, F. & Giri, N., 1990. "On the best equivariant estimator of mean of a multivariate normal population," Journal of Multivariate Analysis, Elsevier, vol. 32(1), pages 1-16, January.
    4. Hendriks, Harrie & Landsman, Zinoviy, 1996. "Asymptotic behavior of sample mean location for manifolds," Statistics & Probability Letters, Elsevier, vol. 26(2), pages 169-178, February.
    5. Hendriks, Harrie, 2005. "The admissibility of the empirical mean location for the matrix von Mises-Fisher family," Journal of Multivariate Analysis, Elsevier, vol. 92(2), pages 454-464, February.
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    Cited by:

    1. Marc Hallin & H Lui & Thomas Verdebout, 2022. "Nonparametric Measure-transportation-based Methods for Directional Data," Working Papers ECARES 2022-18, ULB -- Universite Libre de Bruxelles.

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