CIRJE-F-77. Kuriki, Satoshi and Akimichi Takemura, "Tail probabilities of the limiting null distributions of the Anderson-Stephens statistics", June 2000.

For the purpose of testing the spherical uniformity based on i.i.d. directional data (unit vectors) z_i , i =1,...,n, Anderson and Stephens (1972) proposed testing procedures based on the statistics S_max = max_uS (u) and S _min = min_uS (u), where u is a unit vector and nS (u) is the sum of square of u'z_i's. In this paper we also consider another test statistic S_range = S_max �|S_min. We provide formulas for the P-values of S_max , S_min , S_range 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.