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

✍ Scribed by Satoshi Kuriki; Akimichi Takemura

• Book ID: 104269898
• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 374 KB
• Volume: 89
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1016/s0047-259x(03)00093-9

✦ Synopsis

For the purpose of testing the spherical uniformity based on i.i.d. directional data (unit vectors) z i ; i ¼ 1; y; n; Anderson and Stephens (Biometrika 59 (1972) 613-621) proposed testing procedures based on the statistics S max ¼ max u SðuÞ and S min ¼ min u SðuÞ; where u is a unit vector and nSðuÞ is the sum of squares of u 0 z i 's. In this paper, we also consider another test statistic S range ¼ S max À 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.