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

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.