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) <f>zi, i=1,…,n</f>, Anderson and Stephens (Biometrika 59 (1972) 613–621) proposed testing procedures based on the statistics <f>Smax=maxu S(u)</f> and <f>Smin=minu S(u)</f>, where <f>u</f> is a unit vector and <f>nS(u)</f> is the sum of squares of <f>u′zi</f>''s. In this paper, we also consider another test statistic <f>Srange=Smax−Smin</f>. We provide formulas for the <f>P</f>-values of <f>Smax</f>, <f>Smin</f>, <f>Srange</f> 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. [Copyright &y& Elsevier]