Tests for Randomness of Directions against Equatorial and Biomodal Alternatives
STANFORD UNIV CA DEPT OF STATISTICS
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The paper is concerned with tests of randomness of directions in three-dimensional space or equivalently tests of uniform distribution of points on the unit sphere. One test is against alternatives which concentrate probability density near an equator, and the other test 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 coordinates of theobserved 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 points. Tables of significance are given, based on Monte Carlo studies and the asymptotic distributions, which are derived in the paper. The two-dimensional problem is also discussed.
- Statistics and Probability