Tests for randomness of directions against equatorial and bimodal alternatives
Tests for randomness of directions against equatorial and bimodal alternatives
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针对赤道和双峰替代方案的方向随机性测试
DOI：
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发表时间：
1972
期刊：
影响因子：
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通讯作者：
M. Stephens
中科院分区：
文献类型：
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作者：
T. W. Anderson;M. Stephens
本文讨论了三维空间中方向随机性的检验或单位球上点均匀分布的检验。一个测试是针对那些在赤道附近集中概率密度的替代方案，另一个测试是针对那些在相反的两极附近集中概率密度的替代方案；在每种情况下，极点都是未指定的。测试是基于平方和矩阵的隐根和单位球面上观测点坐标的叉积。对于赤道替代方案，如果最小根小于适当的显著点，则拒绝零假设；对于双峰替代方案，如果最大根大于适当的显著点，则拒绝零假设。根据蒙特卡罗研究和渐近分布，给出了显著点表。对二维问题也进行了讨论。
SUMMARY Tests of randomness of directions in three-dimensional space or equivalently tests of uniform distribution of points on the unit sphere are treated. One test is against alternatives which concentrate probability density near an equator, and the other is against alternatives which concentrate probability density near opposite poles; in each case the poles are unspecified. The tests are based on the latent roots of the matrix of sums of squares and cross-products of the co-ordinates of the observed points on the unit sphere. Against equatorial alternatives the null hypothesis is rejected if the smallest root is less than the appropriate significance point, and against bimodal alternatives the null hypothesis is rejected if the largest root is greater than the appropriate significance point. Tables of significance points are given, based on Monte-Carlo studies and the asymptotic distributions which are derived. The two-dimensional problem is also discussed.