The Annals of Statistics

Nonparametric comparison of mean directions or mean axes

Rudolf Beran and Nicholas I. Fisher

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Samples of directional or axial measurements arise in geophysical, biological and econometric contexts. We represent the rotational difference between two mean directions (or two mean axes) as a direction (or axis). We then construct nonparametric simultaneous confidence sets for all pair-wise rotational differences among the mean directions or mean axes of $s$ samples. By specialization, this methodology yields nonparametric simultaneous tests for pairwise equality of directional means or axes.

Article information

Ann. Statist., Volume 26, Number 2 (1998), 472-493.

First available in Project Euclid: 31 July 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H11: Directional data; spatial statistics
Secondary: 62G15: Tolerance and confidence regions

Rotational difference simultaneous confidence sets simultaneous tests bootstrap


Beran, Rudolf; Fisher, Nicholas I. Nonparametric comparison of mean directions or mean axes. Ann. Statist. 26 (1998), no. 2, 472--493. doi:10.1214/aos/1028144845.

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  • Beran, R. (1979). Exponential models for directional data. Ann. Statist. 7 1162-1178.
  • Beran, R. (1984). Bootstrap methods in statistics. Jahresber. Deutsch. Math. Vereini. 86 212-225.
  • Beran, R. (1990). Refining bootstrap simultaneous confidence sets. J. Amer. Statist. Assoc. 85 417-426.
  • Fisher, N. I. (1985). Spherical medians. J. Roy. Statist. Soc. Ser. B 47 342-348.
  • Fisher, N. I. (1995). Statistical Analy sis of Circular Data. Cambridge Univ. Press.
  • Fisher, N. I., Lewis, T. and Embleton, B. J. J. (1993). Statistical Analy sis of Spherical Data. Cambridge Univ. Press.
  • Fisher, N. I., Lunn, A. D. and Davies, S. J. (1993). Spherical median axes. J. Roy. Statist. Soc. Ser. B 55 117-124.
  • Hall, P. (1986). On the number of bootstrap simulations required to construct a confidence interval. Ann. Statist. 14 1453-1462.
  • Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
  • Kato, T. (1982). A Short Introduction to Perturbation Theory for Linear Operators. Springer, New York.
  • Kent, J. T. (1992). New directions in shape analysis. In The Art of Statistical Science (K. V. Mardia, ed.) 115-127. Wiley, New York.
  • Lewis, T. and Fisher, N. I. (1995). Estimating the angle between the mean directions of two spherical distributions. Austral. J. Statist. 37 179-191.
  • Mardia, K. V. (1972). Statistics of Directional Data. Academic Press, London.
  • Powell, C. McA., Cole, J. P. and Cudahy, T. J. (1985). Megakinking in the Lachlan Fold Belt. J. Struct. Geol. 7 281-300.
  • Watson, G. S. (1983). Statistics on Spheres. Wiley, New York.