The discrimination of mean directions drawn from Fisher distributions

Summary An acceptably accurate approximation for the sampling distribution of the angle between two sample mean directions, conditional on the observed lengths of the vector resultants, is derived for samples drawn from Fisher populations sharing a common true mean direction. From this a test is given for the null hypothesis that two populations (with a common precision parameter) share a common true mean direction. This test is then compared with the unconditional test derived by Watson. The conditional test is then extended to an approximate test for the case where the two populations do not share a common precision parameter. The conditional test for populations with a common precision parameter is then extended to the case where it is desired to test simultaneously whether several samples could have been drawn from populations sharing a common true mean direction. The pooled, unbiased estimate for the inverse of the precision parameter is determined. From this a test for homogeneity of the precision parameter is derived for the case of several samples having unequal sample sizes.