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

Satoshi Kuriki and Akimichi Takemura

Journal of Multivariate Analysis, 2004, vol. 89, issue 2, 261-291

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.

Keywords: Directional; data; Integral; geometry; Maximum; of; a; Gaussian; field; Multivariate; symmetric; normal; distribution; Test; for; spherical; uniformity; Weyl's; tube; formula (search for similar items in EconPapers)
Date: 2004
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