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Smooth Tests of Goodness-of-Fit for Directional and Axial Data

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  • Boulerice, Bernard
  • Ducharme, Gilles R.

Abstract

In this paper we develop, for directional and axial data, smooth tests of goodness-of-fit for rotationally symmetric distributions against general families of embedding alternatives constructed from complete orthonormal bases of functions. These families generalize a proposal of Beran (1979) based on spherical harmonics. Combined with Rao's score test, our alternatives yield simple test strategies. We present a method for constructing an orthonormal basis adapted to the case where the alternatives are first assumed to be rotationally symmetric and then for more general situations. As an example of the versatility of our method, the results are applied to the problem of testing goodness-of-fit for the uniform, the von Mises-Fisher-Langevin, and the Scheiddegger-Dimroth-Watson distributions. It is shown that the proposed test strategy encompasses and generalizes many of the approaches that have so far been proposed for these distributions. Moreover, our method allows for easy adaptation to more complex alternatives than those previously available. In addition, the test statistic can be broken into parts that may be used to detect specific departures from the null hypothesis.

Suggested Citation

  • Boulerice, Bernard & Ducharme, Gilles R., 1997. "Smooth Tests of Goodness-of-Fit for Directional and Axial Data," Journal of Multivariate Analysis, Elsevier, vol. 60(1), pages 154-175, January.
  • Handle: RePEc:eee:jmvana:v:60:y:1997:i:1:p:154-175
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    Cited by:

    1. Figueiredo, Adelaide & Gomes, Paolo, 2006. "Goodness-of-fit methods for the bipolar Watson distribution defined on the hypersphere," Statistics & Probability Letters, Elsevier, vol. 76(2), pages 142-152, January.
    2. Adelaide Figueiredo, 2012. "Goodness-of-fit for a concentrated von Mises-Fisher distribution," Computational Statistics, Springer, vol. 27(1), pages 69-82, March.
    3. Christophe Ley & Thomas Verdebout, 2014. "Skew-rotsymmetric Distributions on Unit Spheres and Related Efficient Inferential Proceedures," Working Papers ECARES ECARES 2014-46, ULB -- Universite Libre de Bruxelles.

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