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 (1972) 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 square 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.