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Tail probabilities of the limiting null distributions of the Anderson-Stephens statistics

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  • Kuriki, Satoshi
  • Takemura, Akimichi

Abstract

For the purpose of testing the spherical uniformity based on i.i.d. directional data (unit vectors) zi, i=1,...,n, Anderson and Stephens (Biometrika 59 (1972) 613-621) proposed testing procedures based on the statistics Smax=maxu S(u) and Smin=minu S(u), where u is a unit vector and nS(u) is the sum of squares of u'zi's. In this paper, we also consider another test statistic Srange=Smax-Smin. We provide formulas for the P-values of Smax, Smin, Srange by approximating tail probabilities of the limiting null distributions by means of the tube method, an integral-geometric approach for evaluating tail probability of the maximum of a Gaussian random field. Monte Carlo simulations for examining the accuracy of the approximation and for the power comparison of the statistics are given.

Suggested Citation

  • Kuriki, Satoshi & Takemura, Akimichi, 2004. "Tail probabilities of the limiting null distributions of the Anderson-Stephens statistics," Journal of Multivariate Analysis, Elsevier, vol. 89(2), pages 261-291, May.
    • Handle: RePEc:eee:jmvana:v:89:y:2004:i:2:p:261-291
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