Abstract

Abstract
As the Watson distribution is much used for modeling axial data, it is important to investigate the existence of a possible outlier in samples from this distribution. So, we generalize to the hypersphere a test of discordancy of an outlier or several outliers en bloc available for the Watson distribution defined on the sphere. We also evaluate the performance of the test for some cases with the usual types of alternatives.

Abstract
The Watson distribution is frequently used for modeling axial data. In this paper, we present goodness-of-fit methods for the bipolar Watson distribution defined on the hypersphere. We analyze by simulation some questions concerning these tests: the adequacy of the asymptotic chi-square distribution used in the tests and the adequacy of using the tabulated critical values for the Kolmogorov-Smirnov statistic when the parameters of the bipolar Watson distribution are unknown. We illustrate these techniques with simulated data from this distribution.

Abstract
The Watson distribution defined on the hypersphere is much used for modeling axial data. This distribution is rotationally symmetric about the modal axis. Then, in practice, before using the Watson distribution for modeling our data, it is better to test the hypothesis of rotational symmetry. For this purpose, we consider the test given by Prentice. In this paper, we determine the empirical power of this test, when data come from a Watson distribution vs. the alternative, where data come from a mixture of two Watson distributions.

Abstract
In multivariate data with n individuals described by p variables it is usual to assume that the variables are fixed and the individuals are randomly selected; in the present study we assume that the individuals are fixed and the previously normalised variables are randomly selected from a population of variables. When we want to associate to the sample of variables a distribution, like the Bingham distribution, it is important to use beforehand a test of uniformity. In this article, we compare the power of Bingham and Gine tests of uniformity for some dimensions of the sphere, vs. a Bingham population or a mixture of Bingham populations.

Abstract
We derive likelihood ratio tests for the equality of the directional parameters of k bipolar Watson distributions defined on the hypersphere with common concentration parameter. We analyze the power of these tests in the case of two distributions supposing in the alternative hypothesis two directional parameters forming an angle, which varies from 18 degrees to 90 degrees. We also compare the likelihood ratio tests with a high-concentration F-test.