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**Sobolev tests of goodness of fit of distributions on compact Riemannian manifolds.**
*(English)*
Zbl 1085.62065

From the introduction: Although many tests of goodness of fit are available for distributions on the circle, comparatively little work has been done on developing general tests of goodness of fit on spheres and other sample spaces used in directional statistics. Goodness-of-fit tests for specific models include score tests for Fisher distributions within the Kent family, Bingham distributions within the Fisher-Bingham family, and for von Mises-Fisher distributions within the Fisher-Bingham family, as well as omnibus tests for Fisher distributions and for Watson distributions. The only general work on goodness-of-fit tests for directional distributions appears to be that of R. Beran [ibid. 7, 1162–1178 (1979; Zbl 0426.62030]] and of B. Boulerice and G. R. Ducharme [J. Multivariate Anal. 60, No. 1, 154–175 (1997; Zbl 0927.62053)]. Beran introduced Wald type tests for certain nested exponential models on spheres, whereas Boulerice and Ducharme considered score tests of goodness of fit of distributions on spheres and projective spaces. Neither Beran’s tests nor those of Boulerice and Ducharme are consistent against all alternatives.

For continuous distributions on the real line or the circle, the probability integral transform can be used to derive a test of goodness of fit from each test of uniformity. However, if the sample space is a manifold of dimension greater than 1, then there is no unique coordinate-invariant analogue of the probability integral transform, so that it is not obvious how one can obtain tests of goodness of fit from tests of uniformity. The purpose of this paper is to use the machinery of M. E. Giné’s [Ann. Stat. 3, 1243–1266 (1975; Zbl 0322.62058)] Sobolev tests of uniformity to obtain coordinate-invariant omnibus tests of goodness of fit on arbitrary compact Riemannian manifolds.

For continuous distributions on the real line or the circle, the probability integral transform can be used to derive a test of goodness of fit from each test of uniformity. However, if the sample space is a manifold of dimension greater than 1, then there is no unique coordinate-invariant analogue of the probability integral transform, so that it is not obvious how one can obtain tests of goodness of fit from tests of uniformity. The purpose of this paper is to use the machinery of M. E. Giné’s [Ann. Stat. 3, 1243–1266 (1975; Zbl 0322.62058)] Sobolev tests of uniformity to obtain coordinate-invariant omnibus tests of goodness of fit on arbitrary compact Riemannian manifolds.

### MSC:

62H11 | Directional data; spatial statistics |

62H15 | Hypothesis testing in multivariate analysis |

62F05 | Asymptotic properties of parametric tests |

### References:

[1] | Beran, R. (1979). Exponential models for directional data. Ann. Statist. 7 1162–1178. · Zbl 0426.62030 · doi:10.1214/aos/1176344838 |

[2] | Best, D. J. and Fisher, N. I. (1986). Goodness-of-fit and discordancy tests for samples from the Watson distribution on the sphere. Austral. J. Statist. 28 13–31. |

[3] | Boulerice, B. and Ducharme, G. R. (1997). Smooth tests of goodness-of-fit for directional and axial data. J. Multivariate Anal. 60 154–175. · Zbl 0927.62053 · doi:10.1006/jmva.1996.1650 |

[4] | Downs, T. D. (1972). Orientation statistics. Biometrika 59 665–676. · Zbl 0269.62027 · doi:10.1093/biomet/59.3.665 |

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[6] | Fisher, N. I. and Best, D. J. (1984). Goodness-of-fit tests for Fisher’s distribution on the sphere. Austral. J. Statist. 26 142–150. |

[7] | Giné, M. E. (1975). Invariant tests for uniformity on compact Riemannian manifolds based on Sobolev norms. Ann. Statist. 3 1243–1266. · Zbl 0322.62058 · doi:10.1214/aos/1176343283 |

[8] | Jupp, P. E. (2001). Modifications of the Rayleigh and Bingham tests for uniformity of directions. J. Multivariate Anal. 77 1–20. · Zbl 1044.62058 · doi:10.1006/jmva.2000.1922 |

[9] | Jupp, P. E. and Spurr, B. D. (1983). Sobolev tests for symmetry of directional data. Ann. Statist. 11 1225–1231. · Zbl 0551.62035 |

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[11] | Kent, J. T. (1982). The Fisher–Bingham distribution on the sphere. J. Roy. Statist. Soc. Ser. B 44 71–80. · Zbl 0485.62015 |

[12] | Khatri, C. G. and Mardia, K. V. (1977). The von Mises–Fisher matrix distribution in orientation statistics. J. Roy. Statist. Soc. Ser. B 39 95–106. · Zbl 0356.62044 |

[13] | Mardia, K. V., Holmes, D. and Kent, J. T. (1984). A goodness-of-fit test for the von Mises–Fisher distribution. J. Roy. Statist. Soc. Ser. B 46 72–78. · Zbl 0543.62043 |

[14] | Mardia, K. V. and Jupp, P. E. (2000). Directional Statistics . Wiley, Chichester. · Zbl 0935.62065 |

[15] | Prentice, M. J. (1978). On invariant tests of uniformity for directions and orientations. Ann. Statist. 6 169–176. · Zbl 0382.62043 · doi:10.1214/aos/1176344075 |

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[17] | Wellner, J. A. (1979). Permutation tests for directional data. Ann. Statist. 7 929–943. · Zbl 0417.62044 · doi:10.1214/aos/1176344779 |

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